IMPROVING THE BEES ALGORITHM FOR COMPLEX … · Algorithm to solve a stochastic optimisation problem. The new modified Bees Algorithm was tested on the optimisation of parameter values

IMPROVING THE BEES ALGORITHM

FOR COMPLEX OPTIMISATION PROBLEMS

A thesis submitted to the Cardiff University

In candidature for the degree of

Doctor of Philosophy

By

Sameh Otri, B.Eng., M.Sc.

Manufacturing Engineering Centre

School of Engineering

Cardiff University

United Kingdom

2011

ii

DECLARATION AND STATEMENTS

DECLARATION

This work has not previously been accepted in substance for any degree and is

not concurrently submitted in candidature for any degree.

Signed ………………………… (candidate) Date ………………

STATEMENT 1

This thesis is being submitted in partial fulfilment of the requirements for the

degree of ………………………… (insert MCh, MD, MPhil, PhD etc, as

appropriate)


STATEMENT 2

This thesis is the result of my own independent work/investigation, except where

otherwise stated.

Other sources are acknowledged by explicit references.


STATEMENT 3

I hereby give consent for my thesis, if accepted, to be available for photocopying

and for inter-library loan, and for the title and summary to be made available to

outside organisations.


STATEMENT 4: PREVIOUSLY APPROVED BAR ON ACCESS

I hereby give consent for my thesis, if accepted, to be available for photocopying

and for inter-library loans after expiry of a bar on access previously approved

by the Graduate Development Committee.


iii

In the name of Allah,

the Most Merciful, the Most Kind

iv

DEDICATION

THIS THESIS IS DEDICATED TO

MY GRANDPARENTS,

MY PARENTS,

MY IN-LAWS,

MY WIFE BAYAN,

AND MY CHILDREN ABDULLAH, JUDE AND JANA.

v

ACKNOWLEDGEMENTS

I would like to express my gratitude to all those who have helped me, in any

way, to successfully complete my PhD thesis.

In particular, I would like to express my sincere thanks to my supervisor, Prof.

D.T. Pham for his encouragement, invaluable advice and guidance throughout

my study.

I owe my parents, brothers and sisters a great gratitude for all their love and

support. Special love goes to my beloved mother, Nadrah, may Allah be mercy

with her.

This work would not have been possible without the support, appreciation and

patience of my wife Bayan and our children Abdullah and Jude.

vi

ABSTRACT

An improved swarm-based optimisation algorithm from the Bees Algorithm

family for solving complex optimisation problems is proposed. Like other Bees

Algorithms, the algorithm performs a form of exploitative local search combined

with random exploratory global search. This thesis details the development and

optimisation of this algorithm and demonstrates its robustness.

The development includes a new method of tuning the Bees Algorithm called

Meta Bees Algorithm and the functionality of the proposed method is compared

to the standard Bees Algorithm and to a range of state-of-the-art optimisation

algorithms.

A new fitness evaluation method has been developed to enable the Bees

Algorithm to solve a stochastic optimisation problem. The new modified Bees

Algorithm was tested on the optimisation of parameter values for the Ant

Colony Optimisation algorithm when solving Travelling Salesman Problems.

Finally, the Bees Algorithm has been adapted and employed to solve complex

combinatorial problems. The algorithm has been combined with two

neighbourhood operators to solve such problems. The performance of the

proposed Bees Algorithm has been tested on a number of travelling salesman

problems, including two problems on printed circuit board assembly machine

sequencing.

vii

CONTENTS

DECLARATION AND STATEMENTS ii

DEDICATION iv

ACKNOWLEDGEMENTS v

ABSTRACT vi

LIST OF FIGURES x

LIST OF TABLES xiii

ABBREVIATIONS xiv

NOMENCLATURE xvi

1. INTRODUCTION 19

1.1 PROBLEM STATEMENT 19 1.2 RESEARCH AIM AND OBJECTIVES 21 1.3 THESIS ORGANISATION 23

2. OPTIMISATION ALGORITHMS AND THEIR APPLICATIONS 25

2.1 OPTIMISATION 25 2.2 OPTIMISATION PROBLEMS 25 2.3 OPTIMISATION ALGORITHMS 26

2.3.1 Neural Networks (NNs) 31 2.3.2 Simulated Annealing (SA) 35 2.3.3 Tabu Search 36 2.3.4 Evolutionary Algorithms 37

2.3.4.1 Evolutionary Strategies 37

2.3.4.2 Evolutionary Programming 38 2.3.4.3 Genetic Algorithms 38

2.3.5 Particle Swarm Optimisation 39 2.3.6 Ant Colony Optimisation (ACO) 40

2.3.6.1 Foraging Behaviour in Ants 40

2.3.6.2 Ant Colony Optimisation Algorithms 41

2.3.6.3 Ant Colony System 42 2.3.6.4 ACO and the Speed of Convergence 49 2.3.6.5 ACS Parameters Tuning 50 2.3.6.6 ACO and Local Search 53

viii

2.3.6.7 Applications and optimisation problems 53 2.3.7 Honeybees Inspiration: Behaviour, Algorithms and Application 54

2.3.7.1 The Nesting Site Selection 54 2.3.7.2 Honeybees Mating Behaviour 56 2.3.7.3 Honeybee teamwork strategy 58

2.3.7.4 Foraging Behaviour of the Honeybees 58 1. BeeHive Algorithm 61 2. BeeAdHoc Algorithm 61 3. The Honey Bee Algorithm 62 4. The Virtual Bee Algorithm 62

5. The ABC algorithm 63 6. Bee Colony Optimisation 64 2.4 THE BEES ALGORITHM AND ITS SUCCESSORS 64

2.4.1 Standard Bees Algorithm 64 2.4.2 Improvements and Applications 64

2.5 SUMMARY 69

2. THE META BEES ALGORITHM FOR SOLVING CONTINUOUS

OPTIMISATION PROBLEMS 70

3.1 A STUDY OF THE STANDARD BEES ALGORITHM PARAMETERS 70 3.1.1 Random Initialisation 71 3.1.2 Neighbourhood / Local Search 76

3.2 THE BEES ALGORITHM PARAMETER TUNING 87 3.2.1 The Meta Bees Algorithm 89

3.3 COMPUTATION EXPERIMENTS 90 3.3.1 The Function Optimisation Problem 90 3.3.2 The Meta Bees Algorithm Implementation 94 3.3.3 Experimental Results 98 3.3.4 Comparison between the Standard Bees Algorithm and the Meta Bees

Algorithm 99 3.3.5 Statistical Analysis of the T-Test 104 3.3.5.1 T-Test Results 106 3.3.5.2 Random Distribution Test 111

3.4 SUMMARY 118

4. MODIFIED BEES ALGORITHM FOR SOLVING STOCHASTIC

OPTIMISATION PROBLEMS 121

4.1 PROPOSED MODIFIED BEES ALGORITHM FOR SOLVING STOCHASTIC

PROBLEMS 122 4.2 COMPUTATION EXPERIMENTS 125

4.2.1 Analysis of the Speed of Convergence of ACS 125 4.2.2 Analysis of Calculating the Average Fitness 134 4.2.3 The Proposed Bees Algorithm 141

4.3 SUMMARY 146

5. APPLICATION OF THE BEES ALGORITHM TO THE SOLUTION OF

COMBINATORIAL OPTIMISATION PROBLEMS 148

ix

5.1 BEES ALGORITHM CHARACTERISTICS FOR COMBINATORIAL OPTIMISATION

PROBLEMS 148 5.2 THE BEES ALGORITHM FOR SOLVING THE TRAVELLING SALESMAN PROBLEM

151 5.2.1 Implementing the Bees Algorithm for the Travelling Salesman Problem 152 5.2.2 The Bees Algorithm with Neighbourhood operators 157 5.2.3 Experiment Results using the Bees Algorithm 163

5.3 THE BEES ALGORITHM AS APPLIED TO THE PRINTED CIRCUIT BOARD

ASSEMBLY OPTIMISATION PROBLEM 166 5.3.1 Printed Circuit Board Assembly 167 5.3.2 The Proposed Bees Algorithm for PCB Benchmark Problems 180 5.3.3 PCB Component Assembly Using a MBTD Machine 183 5.3.4 Experimental Results 185

5.4 DISCUSSION 185 5.5 SUMMARY 194

6. CONCLUSION 195

6.1 CONTRIBUTIONS 195 6.2 CONCLUSIONS 196 6.3 FUTURE WORK 197

REFERENCES 199

x

LIST OF FIGURES

Figure Caption Page

2.1 Topology of an LVQ Network 34

2.2 Application of the ACS for a Simple TSP Problem 48

3.1 Flowchart of the Basic Bees Algorithm 72

3.2 Mean Iteration versus Population Size 73

3.3 Number of Function Evaluation Versus Population Size 74

3.4 Performance for Different Population Sizes 75

3.5 Graphical Explanation of the Neighbourhood Search 78

3.6 Comparison of Different Neighbourhood Strategies for

the De Jong Function

83

3.7 Mean Iteration Required for Different Combination of

Selection

85

3.8 Performance of Different Combinations of the

Selection Method

86

3.9 Pseudocode of the Meta Bees Algorithm 91

3.10 A Bees Algorithm Wraps another Bees Algorithm 93

3.11 Illustration of the Meta Bees Algorithm 96

3.12

(a-e)

Student’s T-Test for both the Bees Algorithm and Meta

Bees Algorithm

109-

110

3.13

(a-b)

Distribution of Random Numbers in both 10 and 100

Bins for the De Jong Function

113

3.14

(a-b)


Bins for the Martin and Gaddy Function

114

3.15

(a-b)


Bins for the Rosenbrock Function

115

3.16

(a-b)


Bins for the Rosenbrock Function

116

xi

3.17

(a-b)


Bins for the Griewangk Function

117

4.1 Pseudocode of the Modified Bees Algorithm for

Solving Stochastic Problems

124

4.2 Fitness Values for the First Experiment over 50 Runs 128

4.3 Oliver30 TSP Problem 129

4.4 Fitness Values for the Second Experiment over 50

Runs

132

4.5 Non-Zero Slope of Regression Line Shows an

Insufficient Number of Evaluations for Calculating the

Average Value

137

4.6 Number of Evaluations - a Representation for 22 Runs 138

4.7 Number of Evaluations - a Representation for 90 Runs 139

4.8 Layers Showing the Bees Algorithm Wrapping the

ACS-TSP

145

5.1 Pseudocode of the Standard Bees Algorithm 156

5.2

(a-b)

A Single-Point Insertion 158

5.3

(a-b)

2-Opt Operator 161-

162

5.4 PCB Assembly Machine of the MBTD Type (with 2

Rotary Turret Heads, 10 Feeder Slots and a Movable

Assembly Table)

177

5.5 Representation of a PCB Assembly Sequence 178

5.6 Flowchart of the Bees Algorithm Solving a PCB

Problem

182

xii

5.7 Evolution of Best Assembly Time for Case 1 – Without

Seeding

189

5.8 Optimal Assembly Sequence (25.92 sec) after 160

Iterations for Case 1 – Without Seeding

192

5.9 Assembly Sequence (29.00 sec) Employed as a Seed

for Case 2 – with Seeding

190

5.10 Evolution of Best Assembly Time (29.00 sec) for Case

2 – with Seeding

191

5.11 Optimal Assembly Sequence (25.08 sec) after 100

Iterations for Case 2 – with Seeding

192

xiii

LIST OF TABLES

Table Caption Page

3.1 Test Functions (Mathur et al., 2000) 82

3.2 The Range of Values Available to the Meta Bees

Algorithm

97

3.3 Standard Bees Algorithm Parameter Settings 100

3.4 Meta Bees Algorithm Parameter Settings 101

3.5 Experimental Results 103

3.6 The Values of Alpha of each Test Function 108

4.1 Parameters employed in the First Experiment 127

4.2 Statistical Information from the First Experiment 130

4.3 Parameters employed in the Second Experiment 131

4.4 Statistical Results from the Second Experiment 133

4.5 Fitness Values for 5, 22 and 90 Runs 140

4.6 Search Spaces of the ACS Parameters 144

4.7 The optimum Results Obtained from the Modified Bees

Algorithm as Compared to Suggested Values in the

Literature

145

5.1 Performance of The Bees Algorithm Compared to GA,

SA and ACS

164

5.2 Types of PCB Assembly Machines and their

Characteristics (Ang et al., 2009)

170

5.3 The Parameters of the MBTD Assembly Machine 184

5.4 The Parameters of the Bees Algorithm 187

5.5 Results Obtained using Different Algorithms 193

xiv

ABBREVIATIONS

2-Opt Two (paths) Optimal

ABC Artificial Bee Colony

ACO Ant Colony Optimisation

ACS Ant Colony System

ATSP Asymmetric Travelling Salesman Problem

CACO Continuous Ant Colony Optimisation

CAP Component Allocation Problem

CIAC Continuous Interacting Ant Colony

CPU Central Process Unit

CSP Component Sequencing Problem

DNA Deoxyribonucleic Acid

EA Evolutionary Algorithms

EP Evolutionary Programming

ES Evolutionary Strategy

FAP Feeder Arrangement Problem

Gas Genetic Algorithms

MBTD Moving Board with Time Delay

MMAS Max-Min Ant Colony System

NE SIMPSA Stochastic Simulated Annealing Optimisation Procedure

PAP Pick and Place Machine

PC Personal Computer

PCB Printed Circuit Board

PID Proportional Integral Derivative

PSO Particle Swarm Optimisation

RAM Random Access Memory

SA Simulated Annealing

SI Swarm Intelligence

SIMPSA Deterministic Simplex Method

xv

SMT Surface Mount Technology

SQP Sequential Quadratic Programming

TS Tabu Search

TSP Travelling Salesman Problem

xvi

NOMENCLATURE

CHAPTER 2

ijp Probability of moving from node i to node j

ij Pheromone intensity of the corresponding link, ( , )i j

ij Desirability of the corresponding link, ( , )i j

Pheromone amplification coefficient

Heuristic amplification coefficient

The global pheromone evaporation parameter

ij The amount of pheromone deposited

q0 An exploitation/exploration parameter

Local pheromone decay parameter

The initial value of the pheromone trails

Lnn The length of tour produced by NNH

Lbest

The length of the global best tour

Tib

The iteration best tour

Cl The length of the candidate list

CHAPTER 3

q Recruitment strategy parameter

f(x) Objective function

N The number of scout bees

M The number of sites selected

E The number of top-rated (elite) sites

Nep The number of bees recruited for the best e sites

0

xvii

Nsp The number of bees recruited for the other (m-e) selected

sites

Ngh The initial size of each patch

fi(X) Objective functions

maxF Maximum value of the function F

minF Minimum value of the function F

X The column vector

CHAPTER 5

Mean value of the process variable being monitored

Standard deviation of the process

d(ci,cj ) The distance between cities i and j

F1 A feeder

H An assembly head

F2 Multiple feeders

DTotal The total distance between all visited cities (components)

V The average speed of movement of the table or assembly

head

TTotal The cycle time

N The total number of components

Ti The time required to place the component Ci

G The number of assembly heads positioned between the pick-

up and placement heads on the turret

t1 The time for the table to move from the location of

component Ci-1

C0 The starting position of the table

xi x coordinate of the component ci

yi y coordinate of the component ci

vx The velocity of the x-y table in the x direction

xviii

vy The velocity of the x-y table in the y direction

t2 The time for the feeder array to change the pick-up feeder

from the feeder supplying component Ci+g to that supplying

component Ci+g+1

Ci+g+1 The component that is to be picked-up when component Ci is

placed onto the PCB

CN The last component to be assembled onto a given PCB

xi The x coordinate of the feeder fi

yi The y coordinate of the feeder fi

vf The speed of the feeder carrier

N The number of cities in a TSP

19

CHAPTER 1

INTRODUCTION 1.1 PROBLEM STATEMENT

The challenges faced by industry today to increase efficiency, especially in

relation to the use of expensive resources within tighter time constraints,

have posed difficult issues for engineers. These challenges have been met

with new approaches to processing embodied in new optimisation

techniques as traditional optimisation techniques are no longer adequate.

New intelligent optimisation algorithms have emerged in the field of

artificial intelligence, many of them inspired by nature. One such algorithm

is the Bees Algorithm which mimics the foraging behaviour of honeybees.

The Bees Algorithm has been initially applied to a range of continuous

problems only. Also, it has not yet been tested for dynamic problems.

Finally, it has a large number of parameters that need to be tuned to produce

good results. However, the tuning normally has to be carried out manually,

which can be a laborious process.

20

The main motivations for the research presented in this thesis were:

1. To improve the Bees Algorithm, enabling it to address the above-

mentioned issues. The improvement includes a new method of

automatically tuning the Bees Algorithm called the Meta Bees Algorithm.

2. To demonstrate the robustness and efficiency of the new algorithm

in comparison to existing algorithms.

3. To ascertain the applicability of the algorithm to challenging

optimisation problems by implementing it for a Travelling Salesman

Problem, a Printed Circuit Board problem and a stochastic optimisation

problem.

21

1.2 RESEARCH AIM AND OBJECTIVES

The overall aim of this work was to prove the hypothesis that the new Bees

Algorithm is able to solve complex optimisation problems faster than other

optimisation techniques.

The following objectives were set to achieve this aim:

1. Survey current swarm-based optimisation algorithms, including the

Bees Algorithm.

2. Develop a new method of evaluating the fitness function of the Bees

Algorithm to accommodate non-deterministic types of problems.

3. Implement new local search operators into the neighbourhood search

process with the aim of reducing the number of parameters needed to

run the Bees Algorithm.

4. Apply the proposed optimisation tools to different categories of

continuous and combinatorial optimisation problems.

5. Validate the different versions of the proposed algorithm by

applying them to different benchmark optimisation problems and

compare the results obtained with those of other optimisation

methods.

22

To achieve the above objectives, the following methodology was adopted:

• Review of previous work: an extensive survey was performed of the state

of the art in swarm-based optimisation techniques, focusing on bees-

inspired algorithms, to identify research trends and potential solutions.

• Algorithm development and evaluation: the standard Bees Algorithm was

extended by adding a new evaluation method for the fitness function of both

local and global search parts of the Bees Algorithm. The performance of the

new version of the algorithm was assessed by running it on a number of

benchmark problems. The results obtained were compared with those of

other optimisation techniques to test the effectiveness of the proposed

method.

23

1.3 THESIS ORGANISATION

The remainder of the thesis is organised as follows: Chapter 2 reviews the

background literature on swarm-based optimisation algorithms relevant to

the work presented in the thesis. This covers material on the Evolutionary

Algorithms (EA), the Genetic Algorithms (GAs), Ant Colony Optimisation

(ACO), Particle Swarm Optimisation (PSO) and bees-inspired algorithms

including the Bees Algorithm itself. A number of recent papers concerning

the optimisation of the Bees Algorithm are also discussed.

Chapter 3 describes a study of the main characteristics of the standard Bees

Algorithm. This is undertaken through an exploration of the parameters of

the algorithm in order to help understand the methods by which its

performance is improved, such as avoiding premature convergence. The

study reveals the implementation of a new method of tuning the Bees

Algorithm called Meta Bees Algorithm (a Bees Algorithm within a Bees

Algorithm). The results of this study were tested with five benchmark

problems and compared with those obtained by other optimisation

algorithms.

Chapter 4 describes the use of the Bees Algorithm to solve a stochastic

optimisation problem using statistical analysis. The algorithm employed to

carry out this task was designed with a new fitness evaluation method based

on computing the average fitness value for each bee over a number of trials

24

rather than computing the value of a single trial. The method enabled the

algorithm to be applied to those situations where the fitness value changes

even when the location visited by a bee remains the same. To test the

algorithm, the parameter value optimisation of a metaheuristic method is

employed. This test took the shape of parameter setting for the Ant Colony

Optimisation algorithm which is employed to solve a Travelling Salesman

Problem (TSP).

Chapter 5 studies the applications of the Bees Algorithm to combinatorial

problems. The development of the Bees Algorithm for the TSP serves as an

illustrative example for such applications and provides a platform to

demonstrate the characteristics of the proposed algorithm. The chapter also

discusses those features of the Bees Algorithm employed in solving more

complex combinatorial optimisation problems and their application to

facilitate the optimisation of a solution to two problems associated with a

Printed Circuit Board (PCB) assembly line, namely, component sequencing

and feeder arrangement.

Chapter 6 summarises the main contributions of this work and the

conclusions reached and provides suggestions for future work.

25

CHAPTER 2

OPTIMISATION ALGORITHMS AND THEIR

APPLICATIONS

This chapter introduces optimisation techniques and algorithms and some

terms and definitions employed in optimisation. It briefly reviews the state

of the art optimisation algorithms employed in this work for comparison

with the Bees Algorithm. The chapter emphasises two main algorithms,

namely those inspired by ants and honeybees. How they function both in

nature and in engineering practice is looked at. The application of these two

algorithms to different types of problems is also considered.

2.1 OPTIMISATION

Optimisation seeks to find the “best” solution to a problem and it also

studies algorithms or methods applied to solve that problem.

2.2 OPTIMISATION PROBLEMS

Each optimisation problem consists of four essential components: an

objective function or fitness function to be optimised, a set of variables that

need to be calculated to find the value of the objective function(s), a set of

constraints that determine the allowed values of the variables, and the search

space that encompass all possible solutions to a problem. With regards to

these four components:

26

1. The degree of nonlinearity of the objective function determines whether

the problem solved is a linear or nonlinear problem. In addition, if the

criteria of the optimisation problem can be expressed in one objective

function it is called a single-objective problem, otherwise, in a multi-

objective problem a number of objective functions are needed.

2. The type of variables employed divide problems into either continuous

problems, or discrete and combinatorial problems must be considered. In

continuous problems the variables employed in the objective function are

real values, whereas in discrete and combinatorial problems they are

restricted to assume only discrete values.

3. If the problem has no constraints or conditions that satisfy it, it is called

an unconstrained problem, otherwise it is called a constrained problem

where it contains one or more constraints that must be satisfied.

4. The search space determines if the problem is a static/deterministic

problem which does not change over time, or if it is a dynamic/stochastic

problem where the search space changes over time (Blackwell and Branke,

2004).

2.3 OPTIMISATION ALGORITHMS

The theory and applications of optimisation algorithms have recently been

developed rapidly in the field of artificial intelligence and the following

27

provides a brief description of the various elements of these algorithms

(Eberhart et al., 2001).

1. On the types of searches applied to solve the optimisation problem there

are two possibilities: Single Point Search (Trajectory) (SPS) which is also

known as a Direct Search (DS), and Population-Based Search (PBS) which

is also known as a Swarm Based Search (SBS). With SPS the algorithm

generates a single solution. Whereas with PBS, a strategy is employed that

generates variations of the tuning parameters. Most search methods use a

greedy criterion to make this decision, which accepts the new parameter if

and only if it reduces the value of the objective or cost function.

2. On the number of solutions generated, there are two possibilities: a

Single Optimum Solution (Single Objective) and a Multi Objective

Optimisation/Multiple Optimal Solution.

3. On the search space of candidate solutions algorithms have employed

two methods: Exploration (Diversification) also known as a Global Search

and Exploitation (Intensification) also known as a Local Search. Local

Search algorithms exploit neighbourhoods while Global Search algorithms

explore the entire search space.

4. On the accuracy of generated solutions (Time/accuracy), three outcomes

are possible (Laporte, 1992):

28

o Exact algorithms are methods which utilise mathematical models and try

to find an optimal solution if it is allowed to complete their execution and

they try to prove that the solution obtained is actually an optimal one.

Unfortunately these types of techniques are time consuming. Some

instances in the exact methods category are Branch and Bound (Wiener

2003) and Integer Linear Programming (Rego and Glover, 2002).

o Approximate/Heuristic Solution: This provides high quality solutions in

short computation time but the algorithm does not guarantee finding an

optimal solution and may fall in local optima missing the true optimum

solution. Approximation algorithms make use of certain heuristics and

iterative improvements to the problem solving process. The approximation

algorithms can be further divided into two groups: constructive heuristics

and improvement heuristics. Instances in constructive heuristics include

Nearest Neighbourhood, Greedy Heuristics and Insertion Heuristics

(Rosenkrantz et al., 1977).

o Metaheuristic algorithms: These are general-purpose techniques for

guiding and modifying problem-specific constrictions or local search

heuristics. They consist of concepts that can be employed to define heuristic

methods and they can be applied to different optimisation problems with

relatively few modifications. They can produce solutions beyond those that

are normally generated in a search for local optimality. Procedures based on

29

evolutionary approaches, Tabu Search, Simulated Annealing, and Multistart

strategies fall into this category. Hybrid procedures based on metaheuristic

frameworks are also considered as metaheuristic algorithms (Glover and

Kochenberger, 2003).

5. In the way the algorithm builds a solution, there are two methods:

o The first is known as constructive, where the algorithm generates

solutions from scratch by adding solution components systematically (step

by step). They are speedy and return reasonably good solutions but not

always guaranteed. An example of this algorithm is the Greedy Constriction

Heuristic (GCH).

o The second is known as Improvement (Local Search), where the

algorithm improves the current solution by movements to neighbouring

solutions. Iterative Improvement Algorithms (IIAs) use this method to find a

better solution and replace the current one. However it may get stuck in a

poor quality local optimum. An example of an IIAs is the Two (two paths)

Optimal (2-Opt) algorithm (Okano et al., 1999) that will be described in

detail in chapter five of this thesis. There are algorithms that employ both

techniques, they initially use the constructive method then follow it up with

the Improvement method such as the Ant Colony Optimisation (ACO).

Instances in improvement heuristics include k-opt (Chandra et al., 1999),

30

Lin-Kernighan Heuristics (Aarts et al., 1988; Helsgaun, 2000), Simulated

Annealing (Knox, 1994), Tabu Search (Freisleben and Merz, 1996),

Evolutionary Algorithms (Merz and Freisleben, 1997; White and Yen,

2004), the Ant Colony System (Stützle and Hoos, 1997) and the Bee System

(Lucic and Teodorovic, 2003; Sato and Hagiwara, 1997).

6. In the method used to improve solutions, algorithms have employed two

techniques:

o Deterministic and Stochastic: with the first approach, random elements

are not employed whereas with the second approach random elements are

employed such as the pheromone value used in the Ant Colony

Optimisation (ACO). This approach will be explained later in the chapter

discussing ACO.

7. Algorithms can also be classed according to where they draw their

inspiration from. Of the most recent are a group of algorithms which get

inspiration from natural systems such as social, ecological, biological,

physical and chemical systems. Two examples of nature-inspired algorithms

are Artificial Intelligence (AI) and Swarm Intelligence (SI) (Bonabeau et al.,

1999).

8. Search History: Memory-less algorithms use the current state of the

search process to determine the next action. Other algorithms incorporate a

31

memory of the past history of the search such as recently performed steps

and the generated solutions.

9. Encoding solutions: Algorithms, like the Bees Algorithm, encode

solutions with real-value variables when solving continuous problems, while

the same algorithm encodes solutions with discrete variables when solving

discrete or combinatorial problems.

After this introduction to essential definitions and terms in the field of

optimisation for both algorithms and problems, a review of current

metaheuristic algorithms is presented.

2.3.1 Neural Networks (NNs)

In the field of Artificial Intelligence (AI), Artificial Neural Networks

(ANNs) are composed of simple interconnecting elements called artificial

neurons. These elements are inspired by biological nervous systems. As in

nature, the network function is determined largely by the connections

(weights) between elements and thus the neural network can be trained to

perform a particular function by adjusting the values of these connections

(Pham and Liu, 1995).

The ANN in general consists of three layers; an input layer connected to a

hidden layer, which is connected to an output layer. Each layer comprises of

32

a set of vectors. The weights on the connection between the input vectors

and the hidden vectors determine the activity of each hidden layer. ANNs

are trained by comparing the output and the target of the network until they

are matched.

Neural networks have been trained to perform complex functions in various

fields of application including pattern recognition, identification,

classification, speech, vision and control systems.

There are many types and structures of ANNs such as Feed-forward neural

networks (Kennedy and Eberhart, 1995; Pham and Sholedolu, 2008),

Spiking Neural Networks (Pham and Sahran, 2006) and Learning Vector

Quantisation (LVQ) (Pham et al., 2006a).

Learning Vector Quantisation (LVQ)

The LVQ neural network was developed by Kohonen (Kohonen, 1989) and

has been successfully employed for many classification problems. Figure

2.1 shows an LVQ network, which consists of three layers of neurons: an

input layer (buffer), a hidden layer and an output layer.

The network is fully connected between the input and hidden layers and

partially connected between the hidden and output layers, with each output

neuron linked to a different cluster of hidden neurons (also known as

33

Kohonen neurons). The weights of the connections between the hidden and

output neurons are fixed at 1. The weights of the input to hidden neuron

connections form the components of "reference" vectors, with one reference

vector assigned to each hidden neuron. When an input vector is supplied to

the network for recognition, the hidden neuron whose reference vector is

closest in terms of Euclidean distance to the input vector is said to win the

competition against all the other hidden neurons to have its output set to "1".

All other hidden neurons are forced to produce a "0". The output neuron

connected to the cluster of hidden neurons that contains the winning neuron

also emits a "1" and all other output neurons, a "0". The output neuron that

produces a "1" gives the class of the input vector, each output neuron being

dedicated to a different class.

The learning method is supervised (Jain and Dubes, 1988) and based on

"competitive" learning, in which neurons compete to have their weights

updated. During learning, the neurons in the hidden layer compete amongst

themselves in order to find the winning neuron whose weight vector is most

similar to the input vector (Kohonen, 1990). The winning neuron gives the

class of the input vector. Only the winning neuron will modify its weights

using a positive or negative reinforcement learning formula, depending on

whether the class indicated by the winning neuron is correct or not. If the

winning neuron belongs to the same class as the input vector (the

classification is correct), it will be allowed to update its weights and move

34

slightly closer to the input vector (positive reinforcement). On the contrary,

if the class of the winning neuron is different from the input vector class

(the classification is not correct), it will be made to move slightly further

from the input vector (negative reinforcement).

Figure 2.1 Topology of an LVQ Network

35

LVQ Network Training Procedure

The training of an LVQ network can be regarded as the minimisation of an

error function. The error function defines the total difference between the

actual output and the desired output of the network over a set of training

patterns (Pham and Oztemel, 1992). Training proceeds by presenting to the

network a pattern of known class taken randomly from the training set. If

the class of the pattern is correctly identified by the network, the error

component associated with that pattern is null. If the pattern is incorrectly

identified, the error component is set to 1.

The procedure is repeated for all the patterns in the training set and the error

components for all the patterns are summed to yield the value of the error

function for an LVQ network with a given set of reference vectors.

2.3.2 Simulated Annealing (SA)

Annealing is the process of heating up a material and then cooling slowly

until it crystallises which allows the metal to achieve a better crystal

structure which is more stable and hard-wearing. Heating up the atoms of

this material increases their energies and these energies give these atoms a

great deal of freedom in their ability to restructure themselves. As the

temperature is reduced the energy of these atoms decreases. Ideally the

temperature should be deceased at a slower rate. If this cooling process is

36

carried out too quickly many irregularities and defects will be seen in the

crystal structure. A slower fall to the lower energy rates will allow a more

consistent crystal structure to form.

Simulated annealing was developed by Scott Kirkpatric in the mid 1970’s to

simulate the actual process of annealing (Pham and Karaboga, 2000).

Simulated annealing begins at a very high temperature where the input

values are allowed to assume a great range of random values. As the

training progresses the temperature is allowed to fall. This restricts the

degree to which the inputs are allowed to vary. This often leads the

simulated annealing algorithm to a better solution.

2.3.3 Tabu Search

Tabu Search (TS) is a metaheuristic algorithm that uses a local or

neighborhood search procedure to iteratively move from a solution x(i) to a

solution x(i+1) in the neighborhood of x(i). TS explores new areas of the

search space by modifying the neighborhood structure of each solution as

the search progresses until some stopping criterion has been satisfied. It uses

a short-term memory structures to determine the new solutions. A Tabu list

contains the solutions that have been visited in the recent n number of

previous solutions. These solutions will be excluded in the search unless one

of these solutions is better than the recently-discovered best solution (Pham

37

and Karaboga, 2000). The TS can be employed for solving combinatorial

optimisation problems, such as the travelling salesman problem (TSP).

2.3.4 Evolutionary Algorithms

Evolutionary Algorithms (EAs) are stochastic search algorithms that are

inspired by the metaphor of natural Darwinian biological evolution. Natural

selection and adaptation in Darwinian evolution are the key sources of

inspiration, driving the EAs candidate solutions towards the optimum by

‘survival of the fittest’. An EA consists of a population of individuals each

having a fitness value, and a genome encoding the main features of the

candidate solution to the given problem. The methods employed in EAs

have been described by several researchers (Michalewicz, 1996 and

Goldberg, 1989). General to all EAs is also a selection pressure mechanism

that removes poor individuals from the population, thus allowing better

individuals to monopolise the evolutionary process. EAs also modify the

individuals to refine the population of candidate solutions.

Four different implementations of the EAs led to the following four

techniques. The following describes the different EAs.

2.3.4.1 Evolutionary Strategies

In the 1960s, Rechenberg and his colleagues were the first to apply the

principle of Darwinian evolution to the study and design of engineering

38

systems and modern technology (Rechenberg, 1965). Standard Evolutionary

Strategies (ES) use a population of individuals then apply mutation,

recombination, and selection operators in order to evolve (evaluation the

evolution) iteratively better and better solutions (Baeck et al., 1991).

2.3.4.2 Evolutionary Programming

Evolutionary Programming (EP) (Fogel et al., 1966) traditionally employed

a representation tree to develop automata recognising strings in formal

languages. However, it was only ten years later that EPs gained worldwide

popularity following the creation of Genetic Algorithms (GAs) by Holland

(1975).

2.3.4.3 Genetic Algorithms

Genetic Algorithms (GAs) are in many ways very similar to Evolution

Strategies (ESs). However, the original applications for which GAs and ESs

were developed are different. While ESs were applied first to continuous

parameter optimisation problems associated with laboratory experiments,

GAs were designed to solve discrete or integer optimisation problems.

In its basic structure, a GA utilises three separate operations for generating a

new solution. These operations are population selection, recombination and

mutation. Candidate solutions in the traditional GA are encoded in binary

bit strings (`chromosomes') for integer and decision variables (Goldberg,

1989). While continuous control variables are approximated and rescaled by

39

equivalent integer variables. A GA initially selects its population randomly

then it applies the principle of `survival of the fittest'. A GA attempts to

construct new improved solutions by combining the features of good

existing ones in a procedure called crossover. To maintain diversity within

the population, a GA runs a mutation operator that changes the bit value, in

the case of a binary coding, from 0 to 1 and vice versa. A GA evaluates the

function(s) of the problem as a fitness value for each member of each

population to assess that particular population. Mutation rate and crossover

rate are two essential parameters required to be tuned carefully.

2.3.5 Particle Swarm Optimisation

The natural flocking and swarming behaviour of birds studied by Craig

Reynolds in the late 80s inspired Russel Ebenhart and James Kennedy to

introduce the PSO algorithm as a recognised and suitable technique

(Kennedy and Eberhart, 1995).

PSO consists of a number of individuals referred to as particles. Each

particle in a PSO has a position and a velocity. These particles are attracted

to positions in the search space that have high fitness. Each particle has a

memory function that remembers two pieces of information, the first piece

of information results from the memory of the particle of its past states as

the best-so-far position that it has visited, called the local best, and the

second piece of information results from the collective experience of all

40

particles as the global best position attained by the whole swarm, called the

global best. Both the local best position of each particle and the global best

position of the entire swarm guide the movements of all particles towards

new improved positions and eventuality to find the global minima/maxima.

PSO has been applied to many problems such as the training of Feed-

forward neural networks (Pham and Sholedolu, 2008) and Clustering

(Omran et at., 2005).

2.3.6 Ant Colony Optimisation (ACO)

ACO is one of the most successful metaheuristic algorithms inspired by the

foraging behaviour of real ant colonies and was proposed by Dorigo and

colleagues for the solution of combinatorial optimisation problems (Dorigo

et al., 1996). The collective trail laying and trail following behaviour of ants

enable them to find the shortest path from the food source to their nest.

2.3.6.1 Foraging Behaviour in Ants

Ants start their journey for food from their nest by exploring the

surrounding area randomly. When food is found, some of it will be carried

back to the nest by the ants which will also lay a chemical substance called

pheromone on the ground while walking back. The amount of pheromone

trails laid by ants recruits more ants to choose the same path to the food

source and guides others to the nest. When an ant deposits pheromone on a

41

path that is not one of the shortest, the pheromone will evaporate. As time

passes the path will not be followed by other ants and this mechanism is

called negative feedback. The more ants that follow a certain path, the more

pheromone will be deposited and this reinforces the quality of this path.

This mechanism is called positive feedback. Hence the ants find the shortest

paths between the nest and food sources (Dorigo and Stützle, 2004;

Deneubourg et al., 1990).

2.3.6.2 Ant Colony Optimisation Algorithms

The Ant Colony Optimisation (ACO) algorithm is a constructive search

algorithm based on the simultaneous exploration of different solutions by a

colony of identical ants. All ant-based algorithms use the positive feedback

mechanism represented in the trail-laying trail-following behaviour of real

ants by reinforcing good solutions or parts of them. The negative feedback

mechanism is implemented through pheromone evaporation to avoid

premature convergence (Stagnation) and being trapped in local optima.

Ant colony optimisation has been studied thoroughly and many algorithms

have been developed such as the Ant System (AS), Max-Min Ant System

(MMAS), Ant Colony System (ACS) and others (Engelbrecht, 2005). The

major differences between these algorithms are:

The way the pheromone update is performed

42

The management of the pheromone trails

The following summarises one of the successful algorithms of the ACO,

namely the Ant Colony System (ACS). This algorithm has been tested and

researched the most.

2.3.6.3 Ant Colony System

The Ant Colony System (ACS) is based on four elements that are employed

to solve the optimisation problem. These elements are an

exploration/exploitation transition rule, a global pheromone trail updating

rule, a local update for the pheromone trail, and the use of a candidate list

(Stützle and Hoos, 1997).

The first element of the ACS is that ants use an exploration/exploitation

decision rule, called the pseudo-random-proportional rule, in which an ant k

located at city i chooses a city j Є k

iN to move to using Equation (2.1)

(Dorigo and Stützle, 2004).

);( )},()({maxarg

); ( ,

0 onExploitatiqqiftt

nExploratioBiasedotherwiseJ

ililkiNl

j

Equation (2.1)

(Dorigo, 2004)

Where ij is the pheromone intensity and it represents the effectiveness of

the move from node i to node j as expressed in the pheromone intensity of

43

the corresponding link, ηij =1/dij is a heuristic value which is the reverse of

the distance between the two cities i and j, β is a parameter that determines

the relative influence of the heuristic value, k

iN is the feasible

neighbourhood of ant i when at city k.

q is a random variable uniformly distributed over [0,1], q0 is a tuneable

parameter (0 ≤ q0 ≤ 1), and J is a random variable selected according to the

probability distribution given by Equation (2.2) (Dorigo, 2004).

kiNl iJiJ

iJiJk

iJtt

tttp

)()(

)()()(

Equation (2.2)

(Dorigo, 2004)

Where is a parameter to control the influence of ij ( is equal to 1 when

q ≤ q0). This decision rule is employed to balance between exploration and

exploitation and has a double function. When q ≤ q0, the decision rule

exploits the knowledge available about the problem such as the heuristic

knowledge about distances between cities in the case of TSP problem, and

the learned knowledge memorised in the form of pheromone trails.

However, when q > q0 it operates a biased exploration of other tours (arcs).

The second element of ACS is where ants perform online step-by-step

pheromone local updates to favour explorations of other new solutions

(cities in case of the TSP) instead of the best current solution. The local

44

pheromone trail updates are performed by applying Equation (2.3) (Dorigo,

2004).

0)()1()( tt ijij

Equation (2.3)

(Dorigo, 2004)

where 0 < ρ ≤ 1 is a parameter governing local pheromone decay. τ0 is the

initial value of the pheromone trails. It was experimentally found that

setting τ0 = (n.Lnn)-1

, where n is the number of cities in the TSP instance and

Lnn is the length of a tour produced by the Nearest Neighbour Heuristic

(NNH), produced good results. The effect of the local update rule is that

each time an ant uses an arc (i, j) its pheromone trail τij is reduced, so that

the arc becomes less desirable for the following ants. This allows an

increase in the exploration of arcs that have not been visited and avoids

stagnation. Stagnation occurs when pheromone accumulates on a certain

path, which is usually a local optimal solution, and more ants keep choosing

this path over and over until eventually all ants choose this path and the

algorithm prematurely converges to this local optimal solution.

The third element of the ACS is the global pheromone trail update where at

the end of an iteration of the algorithm, once all the ants have built a

solution, a pheromone trail is added to the arcs (edges) employed by the ant

that found the best tour from the beginning of the trial. The rule of this

45

offline global pheromone trail update is shown in Equation (2.4) (Dorigo,

2004).

)()()1()1( ttt ijijij

Equation (2.4)

(Dorigo, 2004)

where τij is the pheromone trail level on the edge (ij), φ Є (0, 1) is a

parameter governing the global pheromone decay (evaporation), )(tij =

1/Lbest

where Lbest

is the length of the global-best tour Tgb

found since the

beginning of the trail or the length of the iteration-best tour Tib

found during

the current iteration (t).

With the forth element, the ACS exploits a data structure called the

candidate list that provides additional local heuristic information and

reduces computational time when solving large problems. A candidate list is

a list of preferred cities to be visited from a given city. The candidate list of

a city contains a number of cities ordered by increasing distance. In the ACS

when an ant is in city i, instead of examining all the unvisited neighbours of

city i, it chooses a city to move to that is on the candidate list but has not

been previously visited by that ant. All visited cities are placed on a list

called the Tabu list which is referred to before visiting a city. After visiting

all cities on the candidate list, other cities are then examined.

46

To explain how the ACS operates, the following simple example is

employed where the ACS is applied to solve a TSP which is considered a

minimisation problem and the candidate solution is defined as a sequence of

cities. In this example, there are eight cities (A to H) and for simplicity, it is

assumed that ant one is placed in city A, ant two is placed in city B and so

on. Every time an ant (k) needs to move from city i to city j, it adds its

current location to its Tabu list and then uses Equation (2.1) where it

generates a random value for the parameter q, when q ≤ q0, ant (k) becomes

greedy and exploits the knowledge available about the problem and goes to

city (j) which has the maximum product of the amount of pheromone on the

edge (ij) and the shortest distance between the two cities. While when q >

q0, the ant (k) explores new solutions using a probability decision from

Equation (2.2). Each city has a candidate list; its length is defined by the

number of cities listed (cl). In this example, cl = 2, where cities (B) and (H)

are on the candidate list of city (A) and they will be explored by ant (1)

before other cities. In this example, it is assumed that ant (1) moves from

city (A) to city (B), then city (B) will be added to the Tabu list to avoid

being visited twice by the same ant. After moving from city (A) to city (B),

ant (1) updates the pheromone on the link between the two cities using local

update Equation (2.3). For the next step, ant (1) again calculates the

possibilities of moving from its current city (B) to those cities that are not in

its Tabu list (C to H) using the same Equation (2.1) and so on until ant (1)

47

visits all the seven cities as shown in Figure 2.2. The length of the tour made

by ant (1) will be calculated by adding the length of the arc between each

two cities from the tour. The process will be accomplished by each ant and

at the end of the iteration there will be eight tours generated by eight ants.

The shortest of these tours will be selected as the best tour and the arcs that

form this tour will be updated using the global update formula in Equation

(2.4). The ants are placed again randomly for a second iteration. The

algorithm goes on until a stopping criterion is met such as the minimum

number of iterations or the global tour length has been found.

48

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 2.2 Application of the ACS for a Simple TSP problem

A

D

E

F

G

H

B C

D

A

E

F H

G

B C

D

A

E

F H

G

B C

D

A

E

F H

G

B C

D

A

E

F H

G

B C

D

A

E

F H

G

B C

D

A

E

F H

G

B C

D

A

E

F H

G

49

2.3.6.4 ACO and the Speed of Convergence

This section provides an introduction on the effect of the stochastic nature

of the optimisation problem on the speed of convergence in metaheuristic

methods in general and ACO in particular.

The time and number of evaluations consumed in finding the best solution

in metaheuristic optimisation methods depends on two factors. The first

factor is controllable and the other is not. The first factor is the right

selection of the values of the set of parameters for the metaheuristic method.

The other factor is the stochastic nature of these types of methods i.e. the

randomness that resides in some parts of the algorithm. An example of this

can be seen when the ACS is applied to solve a TSP problem. It is initialised

by setting the value of its parameters such as the number of ants (M) and the

exploitation/exploration ratio (q0). Here, the randomness of the algorithm is

in the distribution of the ants over the available cities. It should be noted

that, for each iteration, these positions are always chosen randomly.

Experiments have been conducted to study the effect of various parameters

on the speed of convergence, here are a summary of these experiments.

- A small size of candidate list (cl) decreases the convergence time

(Dorigo and Gambardella, 1997).

50

- A value of q0 define close to one decreases the convergence time but

the quality of solution is lower (Bonabeau et al., 1999).

- Large numbers of ants increase the solution quality as it widens the

search space but also increase the computation time to a large extent

(Wong and Komarudin, 2008).

- The overall quality of solution is increased when the local

pheromone evaporation rate is close to one but it slows down the

convergence speed and this leads to a suboptimal solution while if it

is close to zero then no cooperative behaviour can emerge (Hao et

al., 2006).

2.3.6.5 ACS Parameters Tuning

Research in parameter setting has provided many ways of approaching the

problem; some researchers, Adenso-Diaz and Laguna (2006) have studied

parameter setting for metaheuristics in general. Their approach is based on a

developed procedure called CALIBRA, however, this approach has the

limitation of tuning only five parameters and the values obtained are not

guaranteed to be the best. (Coy et al., 2001) have provided an approach

based on a statistical design of the experiment and applied it to a vehicle

routing problem. Their method, however, required a rough approximation

based on human experience for the initialisation of the method itself.

Research accomplished by Bartz-Beielstein and Markon (2004) proposes a

51

method to determine relevant parameter setting based again on the statistical

design of experiments and a tree based regression analysis. Birattari (2002)

proposed a procedure that empirically evaluates a set of candidate

configurations by discarding bad ones as soon as statistically sufficient

evidence is gathered against them. It can be noticed from these publications

on general parameter optimisation for metaheuristics that they are based on

statistical analysis.

Other approaches have been carried out to optimise the parameters of ant

systems. These approaches can be divided into three groups; the first group

is that of proposed methods to find the best parameter settings for one or

some of the parameters, the second group proposes experimental analysis to

find the proper parameter set, while the third group tries to connect the

problem instance with the optimal parameters.

Of the first group (Dorigo and Gambardella, 1997) presented a formula for

the optimum number of ants based on the value of ρ and q0. Watanabe and

Matsui (2003) proposed an adaptive control mechanism of the parameter

candidate sets based on the pheromone concentration for improving the

ACO algorithms. Zecchin et al., (2005) developed parametric guidelines for

the application of the ACO to the optimisation of a water distribution

system. Hao et al., (2006) have chosen three parameters to optimise (β, ρ

and q0) and have developed a parameter study strategy based on PSO. A

52

hybridised algorithm using GA and ACS-TSP was attempted by Pilat and

White (2002) to solve the TSP faster but their attempt failed to bring results

better than those of the original ACS (Dorigo and Gambardella, 1997).

Pellegrini et al., (2006) employed the F-Race algorithm to find the four

optimum parameters (m, ρ, α and q0) for the Max-Min Ant Colony System

(MMAS).

Of the second group Pilat and White (2002) proposed a Meta ACS using the

GA as another layer wrapping the ACS to optimise its parameters. They had

better results on their study which considered only three parameters (β, ρ,

q0). Socha (2003) proposed computational studies on some parameters.

Also, Solnon (2002) made computational studies on some parameters of the

ACS as a pre-processing step.

Of the third group Gaertner and Clark (2005) presented a design of an

experiment based on an exhaustive search to find the optimum values of the

three parameters (β, ρ and q0) for a TSP instance. In doing so they tried to

connect the TSP class with the optimum set of parameters. Also, Figlali et

al. (2005) investigated the parameters of the ACS with the randomly

generated job-shop scheduling problem.

53

2.3.6.6 ACO and Local Search

Most ACO algorithms are coupled with operators or local search techniques.

The outcome in performance has been similar to that of the best heuristic

approaches employed (Stützle and Hoos, 1997).

Local search techniques such as 2-Opt or 3-Opt work as improvement

heuristics. When an ACO algorithm provides a feasible tour, local search

techniques repeatedly perform operations (exchanges or moves) which

reduce the tour length until no further improvement is possible (these

operators will be explained in details in chapter 5 section 5.2.2).

2.3.6.7 Applications and optimisation problems

Ant colony optimisation algorithms have been applied to solve a range of

combinatorial (Dorigo et al., 1999) and continuous (Dréo and Siarry, 2004)

optimisation problems. ACO was first employed to tackle combinatorial

problems like the TSP, scheduling problems e.g. the job-shop scheduling

problem (Figlali et al. 2005), the vehicle routing problem (Farooq, 2008),

the quadratic assignment problem (Stützle, 1997) and they were also

employed as classifiers (Martens et al., 2007). The ACO was then employed

for continuous optimisation by Bilchev (Bilchev and Parmee, 1995; Mathur

et al., 2000).

54

2.3.7 Honeybees Inspiration: Behaviour, Algorithms and Application

Honeybees inspired algorithms are a branch of Swarm Intelligence

algorithms which are motivated by the fascinating behaviour of honeybees.

Their behaviour is studied in order to develop metaheuristic algorithms

which can mimic the bees searching abilities. The algorithms are then

employed to find solutions to real life problems. Four forms of honeybee

behaviour have emerged in the literature, namely, the nesting site selection

(Seeley and Visscher, 2003; Passino et al., 2008), the mating behaviour

(Haddad et al., 2006; Sung, 2003), the honeybee teamwork strategy (Sadik

et al., 2006) and the foraging behaviour (Seeley, 1996) and. These types of

behaviour have been modelled to derive various Bees Algorithms with

many applications.

2.3.7.1 The Nesting Site Selection

A swarm of honeybees choosing its future home is one of the most

impressive examples known of an insect group functioning as an adaptive

decision maker. In honeybee nest-site selection (Seeley and Visscher, 2003)

when the size of the hive becomes too small for the honeybees to live in, a

swarm of half the old colony with the mother queen flies a few meters from

the hive and gathers in a tree or on a branch to form a cluster. Then only the

scout bees from this swarm cluster begin to search for potential nest sites in

all directions and at distances of up to several kilometres from the swarm.

55

Scouts assess the quality of sites based on cavity volume, entrance height,

entrance area, and other attributes that are likely correlated with colony

success. A dozen or more potential nest sites are initially advertised by the

returning bees through a representative dance they perform. Eventually the

bees advertise just one site which is not necessarily the one that was first

advertised to the swarm.

During the first stages of this optimisation technique, each returning bee

advertising a site is watched by other ‘‘unemployed’’ scouts which seek to

observe dances. If they easily find a dancer they get recruited to a relatively

high quality site. If they must wait too long to find a dancer, this would

indicate that there are not many good nest-sites currently being assessed so

they explore the environment for more sites. The number of recruits to each

nest-site is in proportion to the number of dances for each site.

At each nest-site there is a quorum-sensing process, where once there are a

certain number of bees at the site, the bees from that site ‘‘choose it’’ by

returning to the cluster to prompt lift-off and then they guide the swarm to

its new home (Beekman et al., 2006). There is significant time–pressure to

complete the nest-site selection process as fast as possible since weather and

energy losses pose significant threats to an exposed colony. However,

enough time must be dedicated to ensure that many bees can conduct

independent evaluations of the site and establish a quorum at a site that is

56

likely to be the best site that the swarm has found. Hence, during nest-site

selection the swarm strikes a balance between time minimisation and site

quality choice maximisation. Agreement among the dancers appears and

within an hour or so of the appearance the swarm lifts off. There is an

increase of dancing just before liftoff. The analysis of the dancing records of

individual scout bees confirmed that there is much turnover in the dancing

bees over the course of a decision making process (Passino et al., 2008).

Most bees that dance for a site cease doing so after a few hours, letting the

next “generation” of dancers carry on the deliberations. Thus it became clear

that a choice of a swarm of a future home is broadly distributed among the

scout bees, and that this leaderless process of group decision-making

consists of friendly competition among the different groups of dancers

representing the different potential nest sites. The groups compete for

additional dancers. Sooner or later, one group of dancers grows numerous

and ultimately excludes its competitors. The site whose dancers prevail in

this winners-take-all contest becomes the new home of the swarm.

2.3.7.2 Honeybees Mating Behaviour

Each normal honeybee colony consists of the queen, drones, workers, and

broods. Queens represent the main reproductive individuals in some types of

honeybees and specialise in laying eggs. Drones are the sires or fathers of

the colony. They are haploid and act to amplify the genome of their mothers

without alteration of their genetic composition except through mutation.

57

Therefore, drones are considered agents that propagate one of the gametes

of their mother and function to enable females to act genetically as males.

Workers specialise in brood care and sometimes lay eggs. Broods arise

either from fertilised or unfertilised eggs. The former represent potential

queens or workers, whereas the latter represent prospective drones.

The marriage process in honeybees was hard to observe as the queens mate

during their mating flight far from the nest. A mating flight starts with a

dance performed by the queens who then start a mating flight during which

the drones follow the queens and mate with them in the air. In a typical

mating-flight, each queen mates with seven to twenty drones. In each

mating, sperm reaches the spermatheca and accumulates there to form the

genetic pool of the colony. Each time a queen lays fertilised eggs, she

retrieves at random a mixture of the sperms accumulated in the spermatheca

to fertilise the egg.

Just as the queen bee is the only bee in a hive that breeds with the others, the

best solution in the pool of solutions is selected to crossbreed with a random

set of others. Thus the algorithm aims to retain the best solutions in the

"gene pool," and achieve a better answer. This differs from the traditional

approach, which selects both parents randomly from the whole pool

(Abbass, 2001). This natural behaviour also led to the Queen-Bee Evolution

method where in a generation the fittest individual crossbreeds with the

58

other bees selected as parents by a selection procedure. This method when

combined with a GA increases the exploitation of the GA. However, it also

increases the probability that the GA will fall into premature convergence

and results in a decrease in the performance of the GA. To decrease the

probability of premature convergence and to reinforce the exploration of a

GA, some individuals in Queen-bee evolution are strongly mutated (Sung,

2003; Azeem and Saad 2004).

2.3.7.3 Honeybee teamwork strategy

In Honeybees, the queen controls the nest and all the other bees provide

various services to the queen. Honeybees move from flower to flower

extracting nectar which they deliver back to the nest where it is employed to

make honey. When a queen dies, a new queen is raised by feeding a normal

worker bee with special food.

Abstract mapping has been done of similarities between this Honeybee

behaviour and agent teamwork strategies, which are later employed in the

design of teamwork architecture and elaborated using prototype case studies

(Sadik et al., 2006).

2.3.7.4 Foraging Behaviour of the Honeybees

Honeybees can exploit a vast number of flower patches by extending their

search over enormous fields surrounding the hive. They search for flower

59

patches that provide plentiful amounts of nectar or pollen that are easy to

collect with less energy usage (Frisch, 1976; Seeley, 1996).

The foraging process during the harvesting season begins in a colony by

employing scout bees to search for adequate flower patches where nectar is

plentiful, easy to extract, and rich in sugar content. Scout bees search

randomly through their journey from one patch to another. Moreover,

during the whole harvesting season, a colony continues its exploration,

keeping a percentage of the whole population as scout bees (Seeley 1996).

When they return to the hive, those who have found a high-quality food

source that is above a certain threshold (a combination of certain

constituents, such as sugar percentage), they deposit their nectar or pollen

that they have collected during the search process and then signal the

position of their discovery to resting nestmates through a ritual known as the

“waggle dance” on the dance floor (Frisch, 1976). The mysterious waggle

dance is essential for colony communication and is performed in a particular

area of the hive called the “dance floor”, and communicates three basic

pieces of information regarding the flower patch: the direction where it is

located, its distance from the hive, and its quality rating (Frisch, 1976;

Camazine et at., 2003). After the waggle dance, the dancer bee goes back to

the flower patch, followed by other nestmates recruited from the hive. The

number of recruited bees depends on the quality rating of the patch. Flower

60

patches that contain rich and easily available nectar or pollen sources attract

the largest number of foragers (Bonabeau, 1998; Seeley, 1996). The

information guides the colony to send its bees to flower patches precisely,

without any supervisory leader or blueprint. The knowledge of each

individual of the outside environment is gleaned solely from the waggle

dance. This dance gives the colony a chance to evaluate different patches

simultaneously in addition to minimising the energy usage rate (Camazine

et al., 2003). This allows the colony to gather food quickly and efficiently.

While harvesting the source, the bees monitor the food level. This

information will be necessary when deciding on the next waggle dance

when they return to the hive (Camazine et al., 2003). If the food source is

still good enough and calls for more recruitment, then that patch will be

advertised by making a waggle dance and recruiting more bees to that

source.

Once a recruited forager returns to the hive, it will in turn waggle dance to

direct other idle bees towards the food source. Thanks to this mechanism,

the most profitable food sources attract the largest number of foragers

(Tereshko and Lee, 2002), and thus the bee colony optimises the efficiency

of the food collection process (i.e. the amount of food collected versus the

cost of harvesting it).

61

During harvesting when the bees detect that there is no more nectar in a

flower patch, bees will abandon it and interrupt the dances intended to

attract other bees to that location.

A number of algorithms has been inspired by bee swarming behaviour and

employed in discrete space, in the next section we will examine several:

1. BeeHive Algorithm

A model borrowing from the principles of bee communication is presented

in (Wedde et al., 2004). The artificial bee agents are employed in packet

switching networks to find suitable paths between nodes by updating the

routing table. Two types of agents are employed – short distance bee agents

which disseminate routing information by travelling within a restricted

number of hops and long distance bee agents which travel to all nodes of the

network. Though the paper talks in terms of bees, it only loosely follows

their natural behaviour.

The BeeHive algorithm (Wedde et al., 2004) was introduced and applied to

routing problems in packet switching networks (Farooq, 2008) where agents

called BeeAgents were employed to route packets among network nodes.

2. BeeAdHoc Algorithm

A new routing algorithm for energy efficient routing in mobile ad hoc

networks was developed by (Wedde et al., 2005) based on the foraging

62

behaviour of honeybees. The algorithm mainly utilises two types of agents,

scouts and foragers, for doing routing in mobile ad hoc networks. The

algorithm, BeeAdHoc, is a reactive source routing algorithm and it

consumes less energy as compared to existing state-of-the art routing

algorithms because in using the principals of foraging behaviour, it utilises

less control packets to do routing. The results showed that the BeeAdHoc

algorithm consumes significantly less energy as compared to state-of-the-art

routing algorithms, without making any compromise on traditional

performance metrics (packet delivery ratio, delay and throughput).

3. The Honey Bee Algorithm

A model generated by studying the allocation of bees to different flower

patches to maximise the nectar intake is described in (Tovey, 2004). This

was subsequently applied to distribute web applications at hosting centres.

4. The Virtual Bee Algorithm

In (Yang, 2005), the author describes a virtual bee algorithm where the

objective function is transformed into virtual food. Unfortunately no

information about how the transformation from objective function to food

source or how agent communication is carried out is given in this work. Nor

are there any comparative results to check the validity of the algorithm.

63

5. The ABC algorithm

For applications in the area of continuous function optimisation, Karaboga

and Basturk (2008) proposed the ABC algorithm. Although the two

algorithms were developed independently, there are strong analogies

between the ABC and the BA. The two optimisation methods can be

described using the same flowchart, and the site abandonment procedure is

also employed in the ABC algorithm. Differently from the BA, the ABC

uses the roulette wheel selection method (Karaboga and Basturk, 2008) to

simulate the recruiting of foragers through the waggle dance. The main

difference between the two algorithms is in the implementation of the local

search procedure. The ABC generates foragers (parents) by a floating point

crossover operation (Pham and Karaboga, 2000) between the dancer bee and

a second bee randomly selected from the population. This operator

calculates the components of the new forager as a weighted average of the

components of the parents. The weight of each component of the parents is

randomly determined. Since the second bee is randomly selected from the

whole population, the crossover operation may generate a forager bee which

is relatively far from the dancer bee. In particular, the forager bee may be

placed outside the fitness peak that it is meant to exploit. For this reason, the

effectiveness of the exploitative search may be reduced, and the extent of

the neighbourhood search is more difficult to control.

64

6. Bee Colony Optimisation

Another implementation of bee behaviour was presented by (Teodorovic et

al., 2006) to solve transportation problems and was called Bee Colony

Optimisation. This algorithm uses a constructive approach that is similar to

ACO.

2.4 THE BEES ALGORITHM AND ITS SUCCESSORS

The Bees Algorithm is a population based search algorithm that imitates the

food foraging behaviour of honeybees to find the optimal solution for both

continuous and combinatorial problem.

2.4.1 Standard Bees Algorithm

The Bees Algorithm balances between the global and the local search. The

BA randomly explores the solution space looking for areas of potential

optimality(s). Then the Bees Algorithm exploits these areas by conducting a

local search, until either a satisfactory solution is found, or a pre-defined

number of iterations has been reached. More detailed explanation of the

Bees Algorithm is discussed in the next chapter.

2.4.2 Improvements and Applications

Various versions of the Bees Algorithm were successfully developed to

solve different engineering problems more efficiently. Their application was

65

also extended to a vast number of new continuous and combinatorial

optimisation problems.

The Bees Algorithm was first applied to solve continuous function

optimisation (Pham et al., 2006b). The Bees Algorithm was tested on a

range of well-known benchmark function optimisation problems of different

degrees of complexity and the experimental results proved the reliability of

the Bees Algorithm (Pham and Castellani, 2009).

The Bees Algorithm was also applied to train different types of neural

network such as the training of the Learning Vector Quantisation networks

(Pham et al., 2006a), the training of Multi-Layered Perceptrons neural

network (Pham et al., 2006c; Pham et al., 2006e), the training of the Radial

Basis Functions network (Pham et al., 2006d), the training of the Spiking

Neural Networks (Pham and Sahran, 2006) and the training of the Support

Vector Machine (Pham et al., 2007a). These trainings were for recognising

patterns in control charts employed for identifying wood defects. In general,

results showed the testing and training accuracies using the Bees Algorithm

as a classifier were either higher than or very close to those accuracies

produced by other classifiers.

Pareto multi-objective optimisation was another extension of the Bees

Algorithm applied to the welded beam design problem. The objective of the

66

design is to minimise the cost of fabrication while finding a feasible

combination of weld thickness, weld length, beam thickness and beam

width. The Bees Algorithm produced better results (less cost) than almost

all the examined algorithms (Ghanbarzadeh, 2007).

The Bees Algorithm was also applied to environmental/economic power

dispatch problems with weighted-sum multi-objective optimisation (Lee and

Haj Darwish, 2008) and with Pareto optimality (Pham et al., 2008a).

A new formulation of the Bees Algorithm was proposed for solving a

chemical engineering process as a dynamic optimisation problem. It

includes new search operators, and a new selection procedure that enhances

the survival probability of newly formed individuals. The proposed

algorithm was tested on six benchmark dynamic optimisation problems. For

all the problems, the Bees Algorithm found very satisfactory optima and the

results proved the high reliability of the proposed technique (Pham et al.,

2008b).

In a biological application involving protein structures, the Bees Algorithm

was adapted to search the protein conformational search space to find the

lowest free energy conformation (Bahamish et al., 2008). Proteins perform

many biological functions in the human body. The structure of the protein

determines its function. In order to predict the protein structure

67

computationally, the protein must be properly represented. The algorithm

was able to find the lowest free energy conformation.

One of the first implementations of the Bees Algorithm in discrete space for

combinatorial problems was in designing cellular manufacturing systems

where the Bees Algorithm was employed for solving the cell formation

problem (Pham et al., 2007b). Experimental results indicated that the Bees

Algorithm is very effective for large-scale problems.

Another application of the Bees Algorithm is the scheduling of jobs with a

common due date for a machine to minimise the penalties associated with

early or late completion (Pham et al., 2007c). Results proved it to be more

stable and robust than other existing optimisation techniques such as GA,

PSO and TS.

The Bees Algorithm was successfully applied for PCB assembly planning

(Pham et al., 2007d). The computational experiments showed that the Bees

Algorithm gives a significant reduction in assembly time compared to the

results obtained with the GA and EP on a benchmark assembly task. The

Bees Algorithm was also copped with TRIZ-inspired operators for the same

application (Ang et. al, 2009) where a shorter time was obtained as

compared to previous work.

68

The Bees Algorithm was also applied to solve the Scholar Timetabling

Problem. Experimental results showed promising results (Lara et al., 2008).

Preliminary design is another application of the Bees Algorithm (Pham et

al., 2007e). The algorithm has been also employed to design mechanical

components (Pham et al., 2008c; Pham et al., 2009).

The algorithm was also employed to obtain the optimal sink path for large-

scale sensor networks (Saad et al., 2008). Another usage of the Bees

Algorithm was in robotics. It was employed for learning the inverse

kinematics of a robot manipulator (Pham et al., 2008e).

The Bees Algorithm was also applied to clustering problems to improve the

results of the K-means (Pham et al., 2007f) and the C-means (Pham et al.,

2008f) algorithms

Interpolation and extrapolation operators were introduced to unselected bees

in the Bees Algorithm by (Ghanbarzadeh, 2007) to improve these bees by

mating them with the selected ones. Also, two new methods were proposed

for the Bees Algorithm namely the ‘shrinking’ method for neighbourhood

size and the idea of ‘abandon sites’ which is employed when stuck in a local

optimum or when no new information is found.

69

In another arrangement, an adaptive neighbourhood search and random

particles were added to the global search by proposing a hybrid PSO-Bees

Algorithm to solve the problem of premature convergence in the basic PSO

algorithm (Pham and Sholedolu, 2008).

2.5 SUMMARY

Various optimisation techniques and algorithms were briefly introduced in

this chapter alongside the terms and definitions often employed in the study

of optimisation. The natural inspiration of these algorithms and their

application to the field of engineering were discussed. Applications of these

algorithms to different type of problems were also examined.

70

CHAPTER 3

THE META BEES ALGORITHM FOR SOLVING

CONTINUOUS OPTIMISATION PROBLEMS

This chapter describes the main characteristics of the standard Bees

Algorithm and a study to explore the parameters of the algorithm to help

understand the methods by which its performance is improved, such as

those employed to avoid premature convergence. The chapter presents a

new method of tuning the Bees Algorithm called Meta Bees Algorithm (a

Bees Algorithm within a Bees Algorithm). The tuned Bees Algorithm has

been applied to a range of function optimisation problems. The results

obtained have been compared against those produced by other optimisation

algorithms.

3.1 A STUDY OF THE STANDARD BEES ALGORITHM

PARAMETERS

As mentioned in the previous chapter, the Bees Algorithm takes its

inspiration from the food foraging behaviour of honeybees to search for the

best solution to a given optimisation problem. As shown in Figure 3.1, the

algorithm randomly samples the solution space looking for areas of high

performance. Throughout the search it performs an exploitative

neighbourhood search combined with a random explorative search. In order

to study these two main searches performed by the Bees Algorithm, a set of

experiments was run where the Bees Algorithm was applied to several

71

function optimisation problems. The effects of the Bees Algorithm

parameters are then discussed.

3.1.1 Random Initialisation

The bee population size is fixed to n scout bees. These bees randomly

sample the solution space with uniform probability across the space. Each

scout bee evaluates the visited site (i.e. solution) via the fitness function.

The population size n is one of the key parameters in the Bees Algorithm.

The effect of changing the population size on the mean number of iterations

to arrive at the correct answer is shown in Figure 3.2. The variation of

population size against the number of evaluated points to arrive at the

correct solution is shown in Figure 3.3. Also shown in Figure 3.4 is a graph

demonstrating the performance of the algorithm with increasing population

size.

Figure 3.2 shows that the number of iterations required to obtain the

solution reduces with increasing population size.

72

Random (n-m)

Fitness Evaluation

Random Initialisation (n)

Solution

Fitness Evaluation

Local Search

Global Search

New Population

Converged?

Selection n

Best Sites m

nbp Bees per Patch

Elite Sites e nep Bees per Patch

Fitness Evaluation

Fitness Evaluation

Select Patch Fittest Select Patch Fittest

Figure 3.1 Flowchart of the Standard Bees Algorithm

Yes

No

73

Population (n)

460

430

400

370

340

310

280

250

220

190

160

130

100

70

40

Me

an

Ite

ratio

ns

500

400

300

200

100

0

Figure 3.2 Mean Iteration versus Population Size

74

Population (n)

460

430

400

370

340

310

280

250

220

190

160

130

100

70

40

Me

an

Ge

ne

rate

d P

oin

ts

40000

30000

20000

10000

Figure 3.3 Mean Number of Function Evaluations versus Population Size

75

Figure 3.4 Performance for Different Population Sizes

Successfulness

0

20

40

60

80

100

0 50 100 150 200

n ( Population )

Fail

s i

n 1

000 r

un

s

76

Figure 3.3 shows how an increase in population size leads to an increase in the

number of function evaluations, which is predictable. Here, the mean generated

points is defined as the mean of the number of times the objective function was

called.

To achieve higher algorithm reliability, a minimum size of population is

required as shown in Figure 3.4 (where Fails represents a case where the

optimum solution of the optimisation problem could not be reached). In order to

arrive at the solution with fewer iterations, the population should be larger than

the population used. To obtain a reasonable number of function evaluations, the

population has to be as small as possible. Within these three constituent parts

(the algorithm reliability, the number of iterations and the number of function

evaluations) a range needs to be set to choose a proper population size.

3.1.2 Neighbourhood / Local Search

As in all the evolutionary algorithms, the neighbourhood search is an essential

concept of the Bees Algorithm. After ranking the sampled solutions and

locating the most promising ones (i.e. the highest ranking locations), other bees

77

are recruited to search the fitness landscape in the neighbourhood of these

solutions.

The neighbourhood search is based on a random distribution of bees in a

predefined neighbourhood range. For each selected site, bees are randomly

distributed to find a better solution. As shown in Figure 3.5, only the best bee is

chosen to advertise its source after which the centre of the neighbourhood field

is shifted to the position of the best bee (i.e. from A to B).

In undertaking the study of this form of local search, three issues have been

taken into account; the number of recruited bees in the neighbourhood range,

the width of range and the method of site selection.

(A) The Number of Recruited Bees

The number of recruited bees around selected sites should be defined properly.

When the number is increased, the number of function evaluations will also be

increased and vice versa, when the number decreases, the chance of finding a

good solution decreases.

78

Figure 3.5 Graphical Explanation of the Neighbourhood Search

* *

*

*

*

Best Bee

* *

*

*

*

A A

B

Fitness Fitness

x x

79

(B) The Patch Size (Neighbourhood Range)

When the neighbourhood range can be arranged adequately, the number of

recruited bees will depend on the complexity of a solution space. The

neighbourhood range is a variable which needs to be tuned for different types of

problem spaces. Three different strategies have been applied to improve the

efficiency and robustness of the Bees Algorithm. These are (1) a fixed

neighbourhood region width strategy, (2) a region changing according to

iteration strategy and (3) hybrid strategies combining the two previous

strategies where, for instance, the first strategy can be employed up to the 50th

iteration and then the second strategy employed up to the stopping criteria.

With the first strategy, a neighbourhood width for all selected sites was fixed to

a certain range that is sufficient to deal with the complexity of the problem

space. In this strategy, beginning from the first iteration, all the bees harvest on

the same size fields which are defined as (ngh) in Equation (3.1).

Neighbourhood_Range = ngh Equation (3.1)

However, in the second strategy, the region changes proportional to the number

of iterations. All sites have the same range value and this range will be

80

narrowed down depending on the iteration as shown in Equation (3.2). This

strategy has been established to increase the accuracy of the solution.

Neighbourhood_Range = (ngh / iteration) Equation (3.2)

Finally, the third strategy was implemented as a combination of the first two

strategies to improve their efficiency.

These neighbourhood range strategies have been tested using the De Jong

function benchmark presented in Table 3.1. These tests were run independently

10 times and the mean of these 10 runs is presented in Figure 3.6 with the

number of iterations being set to 1000. The parameters set for this test were as

follows:

The population is set to 15, the selected sites are 5, the elite sites 2, bees around

elite points numbered 4, bees around selected points numbered 2. A

neighbourhood spread of 0.01 is defined as an initial range for all strategies. For

the third, mixed, strategy the first 20 iterations of the algorithm employed the

first strategy (fixed ranges) and thereafter the second strategy of narrowing

down the ranges was employed. It is clear from the results in Figure 3.6 that the

first strategy reaches its minimum value before any of the other strategies.

81

However, the second and third strategies gave similar results for this relatively

simple problem space. Thus, it does not make sense to employ the more

complex second and third methods as the first strategy can be said to be both a

simple and efficient method for neighbourhood searching.

82

Table 3.1 Test Functions (Mathur et al., 2000)

No Function

Name Interval Function

Global

Optimum

1 De Jong [-2.048, 2.048] 2

1

2

2

2

1)1()(100)93.3905(max xxxF

X(1,1)

F=3905.93

2 Martin and

Gaddy [0, 10]

2

212

21 3

)10()(min

xx

xxF X(5,5)

F=0

3 Rosenbrock

(2D) [-1.2, 1.2] 2

1

2

2

2

1)1()(100min xxxF X(1,1)

F=0

4 Rosenbrock

(2D) [-10, 10] 2

1

2

2

2

1)1()(100min xxxF X(1,1)

F=0

5 Griewank [-600, 600] 1

100cos100

4000

1min

50

1

50

1

2

i

i

i

ii

xxF

X( 001

)

F=0

83

Figure 3.6 Comparison of Different Neighbourhood Strategies for the De Jong

Function

3902

3902.5

3903

3903.5

3904

3904.5

3905

3905.5

3906

3906.5

0 10 20 30 40 50

Iteration

Fit

ness 1st Strategy

2nd Strategy

3rd Strategy

84

(C) The Recruitment Strategy (Site Selection)

Two different techniques were implemented: Probabilistic Selection and Best

Site Selection.

With the Probabilistic Selection technique, the roulette wheel method is

employed and sites with better fitness have more chance of being selected.

However, with the Best Selection technique (greedy selection), the best sites

according to fitness will be selected. Different combinations of the selection

methods, ranging from pure Probabilistic Selection (q=0) to pure Best Selection

(q=1), have been investigated. The mean number of iterations required to reach

the answer and the success of each combination are shown in Figures 3.7 and

3.8.

From the experimental results, the Best Site selection technique demonstrated

higher success. It is simpler to implement as is does not involve the use of a

roulette wheel which causes the algorithm to take longer and makes it more

complicated, and thus the Best Site selection has been recommended for use.

85

Figure 3.7 Mean Iterations required for Different Combinations of methods

86

Figure 3.8 Performance of Different Combinations of the Selection Method

Successfulness

0

20

40

60

80

100

0 50 100 150 200

n ( Population )

Fail

s i

n 1

000 r

un

s

q=0

q=0.1

q=0.5

q=0.9

q=1

87

3.2 THE BEES ALGORITHM PARAMETER TUNING

The Bees Algorithm, like all other metaheuristic search algorithms including

Tabu Search (TS), Simulated Annealing (SA) and ACO, invariably requires a

set of parameters in order to solve complex optimisation problems. These

parameters directly impact on the performance of the solver and as such, the

researcher will often hand-tune parameter values before the application of the

Metaheuristic or use a standard set of values that have been found to be

traditionally well-suited by other researchers.

As shown in the first section of this chapter, the search strategy of the Bees

Algorithm combines global random exploration with local neighbourhood

sampling. The explorative search (scout bees) and exploitative search (recruited

foragers) are clearly differentiated and they can be independently varied

through a set of learning parameters. This clear decoupling between exploration

and exploitation facilitates the tuning of the algorithm.

In spite of the fact that most of the work that has been carried out in this field

states the need for a mechanism to tune the Bees Algorithm parameters,

comparatively very little research has been carried out into the analysis of the

parameter values or the way they can be automatically derived or tuned. The

optimisation of the Bees Algorithm has been carried out, in the past, according

88

to experimental trial and error and the values of its parameters were decided

empirically. Once the learning parameters were manually optimised by

conducting a number of trials, they were fixed and kept unchanged for all the

optimisation problems. A trial to understand the effect of the parameters of the

Bees Algorithm on its performance and the speed of convergence had been

carried out (Pham et al., 2005). The experiments were conducted to test

different parameters of the algorithm and the effects of changing one parameter

while keeping other parameters fixed. Researchers (Pham et al., 2006b) have

indicated that one of the drawbacks of the original Bees Algorithm is the

number of tuneable parameters used and that further work should target the

reduction of the number of learning parameters. There was a trial conducted by

(Pham et al., 2007d) to drop these numbers which concluded that even though

the performance of the Bees Algorithm is fairly robust to reasonable variations

of the learning parameters, a smaller parameter set would ease the optimisation

of the algorithm performance.

No research has been conducted into either the statistical analysis of the Bees

Algorithm parameter values or the way they can be automatically derived or

tuned by metaheuristic algorithms themselves. However, it is possible to run the

Bees Algorithm on top of another Bees Algorithm during the parameter search

89

process and find the optimal settings, although this may incur a computational

overhead.

The proposed combined Bees Algorithm, called Meta Bees Algorithm, is used

to evolve suitable parameter values using its own optimisation process while

solving complex problems.

3.2.1 The Meta Bees Algorithm

In this chapter a technique employed by the Bees Algorithm allows it to self-

adapt its own parameters to minimise the sensitivity of these parameters by

finding an area where the effects of these parameters on the algorithm are less.

The approach adopted uses a standard mechanism of the Bees Algorithm to

modify and determine the appropriate parameter values while the problem is

being solved. Therefore it is conceptually simple to integrate this approach into

the standard Bees Algorithm.

For this work, the Bees Algorithm was ‘wrapped around’ a second Bees

Algorithm (the Wrapper). Figure 3.9 shows the pseudocode of the Meta Bees

Algorithm and Figure 3.10 demonstrates the flow of information between the

two Bees Algorithms. It can be seen that each bee of the Wrapper Bees

Algorithm represents a set of Bees Algorithm parameters and, again, the fitness

90

values are the total number of function evaluations required by the Bees

Algorithm to find the optimum solution for the function optimisation. The Bees

Algorithm uses the same mechanisms for generating solutions to evolve

appropriate values for its parameters.

3.3 COMPUTATION EXPERIMENTS

3.3.1 The Function Optimisation Problem

The performance of the Meta Bees Algorithm was evaluated on a set of five

continuous function minimisation benchmarks (Mathur et al., 2000). Using the

proposed method for each problem, the results obtained were compared with

the results given by the standard Bees Algorithm an with the other optimisation

algorithms. The function minimisation problems represent a varied set of

learning scenarios that were chosen from amongst widely used benchmarks in

the literature (Mathur et al., 2000). Table 3.1 shows the equations of the five

continuous function minimisation benchmarks. For each function, the equation

is given together with the range of the variables and the global minimum. The

Martin and Gaddy benchmark is a fairly simple unimodal function. The

Rosenbrock benchmark is unimodal, the minimum lies at the bottom of a long,

narrow, parabolic shaped valley. Finding the valley is trivial, however locating

the minimum is difficult. The Griewank function has an overall unimodal

91

behaviour, with a rough multi-modal surface created by a cosinusoidal “noise”

component.

1- Initialise the Bees Algorithm Wrapped population with random parameter

values from the Wrapper Bees Algorithm

2- Evaluate fitness of the population of the Wrapped Bees Algorithm.

3- Select the solutions that satisfy the criterion and add them to the solution set

4- While (stopping criterion not met)

//Forming new population.

5- Select sites for neighbourhood search.

6- Determine the patch size.

7- Recruit bees for selected sites (more bees for best e sites) and evaluate

fitness.

8- Select the fittest bee from each patch.

9- Assign remaining bees to search randomly and evaluate their fitness.

10- End While.

Figure 3.9 Pseudocode of the Meta Bees Algorithm

92

93

Figure 3.10 A Bees Algorithm Wraps another Bees Algorithm

The Bees Algorithm - Wrapper

The Bees Algorithm - Wrapped

Function Optimisation Problem

Fitn

ess (No

. of E

valu

ation

s)

Sen

d B

ees (BA

param

eters)

Sen

d B

ees (Dim

ensio

ns)

Fitn

ess (Fu

nctio

n V

alue)

94

3.3.2 The Meta Bees Algorithm Implementation

The Bees Algorithm is characterised by a number of core parameters which are:

the number of scout bees, n, the number of high-quality sites that are selected

for neighbourhood search, m, the number of elite (top-rated) sites amongst the

best m sites, e, the number of bees recruited for a neighbourhood search around

the e elite sites, nep, the number of bees recruited for a neighbourhood search

around the remaining (m-e) sites, nsp, the initial size of each flower patch, ngh,

and the stopping criterion.

In addition to the above seven main parameters, two extra parameters have been

introduced by Ghanbarzadeh (2007) in his thesis; the neighbourhood shrinking

parameter and the site abandonment parameter. However, these two extra

parameters have not been included in this study as the aim of the research was

to show how well the standard Bees Algorithm can perform without the need to

introduce extra parameters.

At each iteration, the standard Bees Algorithm is augmented by allowing each

bee to select a value for each parameter before commencing the selection of the

solution components. Thus each bee maintains its own parameter values and in

turn uses these to adapt the parameter values. Selection of a parameter value is

95

based exclusively on the fitness of the solution i.e. if an improvement in the

fitness is found by a particular value of a set of parameters, the search exploits

around this value to get a better result and so on.

As shown in Figures 3.10 and 3.11, the Meta Bees Algorithm consists of two

Bees Algorithms namely the Wrapper Bees Algorithm and the Wrapped Bees

Algorithm. The parameters values of the Wrapper Bees Algorithm have been

setup empirically by using a pilot test and these are; n1 = 10, m1 = 3, e1 = 1,

nep1 = 5, nsp1 = 3, ngh1 = 0.1 and the stopping criterion is 10000 iterations or

50000 numbers of evaluations.

Each parameter of the Wrapped Bees Algorithm of the Meta Bees Algorithm is

given a suitable range in which its value can lay (Table 3.2). The initial value of

each parameter is randomly chosen within the range.

Every time a value needs to be assigned to a parameter, a point will be

randomly generated within the range given in the table (3.2) with the following

two constrains (Equations 3.3 and 3.4)

m ≤ n Equation (3.3)

e ≤ m Equation (3.4)

96

Bees Algorithm

‘Wrapper’

Bees Algorithm

‘Wrapped’

Function

Optimisation

Number of Evaluations

Fitness Function

n1

m1

e1

nep1

nsp1

ngh1

n2

m2

e2

nep2

nsp2

ngh2

x1

C

x3

x4

B

xd

Figure 3.11 Illustration of the Meta Bees Algorithm

97

Table 3.2 Range of Values Available to the wrapped Bees Algorithm

Bees Algorithm Parameters Symbol

Value

Minimum Maximum

Number of scout bees n 3 20

Number of best selected sites m 3 10

Number of elite sites amongst the best m sites e 1 10

Number of bees recruited for neighbourhood

search around the e elite sites

nep 1 10

Number of bees recruited for best m sites nsp 1 5

The initial size of each flower patch ngh 0.01 0.2

98

3.3.3 Experimental Results

The experiments were performed using the Meta Bees Algorithm to evolve its

own parameter values. It was run 100 times for each parameter setting on each

benchmark problem. For each of the 100 trials, the Wrapped Bees Algorithm

F

50000 learning cycles had elapsed. The number of evaluations was recorded.

The final accuracy result (E) was found to be:

elsexF

xFE

f

f 001.00 Equation (3.5)

Where xf is the final solution generated by the algorithm.

The computing platform used to perform the experiments was a 2.00GHz

Intel(R) Core(TM) 2 Dual CPU PC with 1.99 GB of RAM. The experimental

programs were coded in the C language and compiled with Microsoft Visual

C++. Each problem instance was run across 100 random seeds.

99

3.3.4 Comparison between the Standard Bees Algorithm and the Meta

Bees Algorithm

Table 3.3 shows the empirically derived Bees Algorithm parameter values used

with the different test functions (Pham et al., 2006b).

Table 3.4 shows the results of running the Meta Bees Algorithm to optimise the

parameter values employed in the Wrapped Bees Algorithm while solving the

minimum function optimisation problem.

The characteristic values of each parameter are summarised as follows:

- n: The values were generally between 3 and 7, with 4 being most common.

- m: The value was the same (3) for all the optimisation functions.

- e: The value was the same (1) for all the optimisation functions.

- nsp: The value was the same (1) for all the optimisation functions.

- nep: The range of values was between 7 and 8, with 8 being the more

common.

- ngh: Generally the range of values was between 0.02 and 0.06 with 0.02

being the most common.

100

Table 3.3 Standard Bees Algorithm Parameter Settings

Function No. n m e nsp nep

ngh

(initial)

1 De Jong 10 3 1 2 4 0.1

2 Martin and Gaddy 20 3 1 1 10 0.5

3 Rosenbrock (2D) 10 3 1 2 4 0.1

4 Rosenbrock (2D) 6 3 1 1 4 0.5

5 Griewank 50 5 2 10 20 5

101

Table 3.4 Meta Bees Algorithm Parameter settings

Function n m e nsp nep ngh

1 De Jong 4 3 1 1 8 0.03

2 Martin and Gaddy 4 3 1 1 8 0.02

3 Rosenbrock (2D) 7 3 1 1 7 0.04

4 Rosenbrock (2D) 4 3 1 1 8 0.02

5 Griewank 3 3 1 1 8 0.06

Most common value 4 3 1 1 8 0.02

102

Table 3.5 shows a comparison of the results obtained with the Meta Bees

Algorithm and those obtained using the standard Bees Algorithm

(Ghanbarzadeh, 2007), the Deterministic Simplex Method (SIMPSA) (Mathur

et al., 2000), the Stochastic Simulated Annealing Optimisation Procedure (NE

SIMPSA) (Mathur et al., 2000), the Genetic Algorithm (GA) (Mathur et al.,

2000) and the Ant Colony System (ACS) (Mathur et al., 2000).

The average number of evolutions i.e. the numbers of points visited is shown

for 100 independent runs in Table 3.5. Inspection of the runtimes (number of

evaluations) for both algorithms, the Meta Bees Algorithm and standard Bees

Algorithm, reveal that overall, the Meta Bees Algorithm produced more stable

values for the Bees Algorithm parameters with a smaller number of evolutions

for most of the minimisation functions except the second function (Martin and

Gaddy).

A T-Test and Random Distribution test were performed to evaluate the results

obtained using the standard Bees Algorithm and Meta Bees Algorithm to see if

their means are statistically different from each other. These were to ensure

rigorous statements could be made regarding each set of results.

103

Table 3.5 Experimental Results

Funct. no.

SIMPSA NE-SIMPSA GA ACS

Standard Bees

Algorithm (*)

Meta Bees Algorithm

succ

% Mean no.

of evaluations

succ

% Mean no.

of evaluations

succ

% Mean no.

of evaluations

succ

% Mean no.

of evaluations

succ

% Mean no.

of evaluations

succ

% Mean no.

of evaluations

1 *** *** *** *** 100 10160 100 6000 100 868 100 683

2 *** *** *** *** 100 2844 100 1688 100 526 100 608

3 100 10780 100 4508 100 10212 100 6842 100 631 100 571

4 100 12500 100 5007 *** *** 100 7505 100 2306 100 1471

5 *** *** *** *** 100 200000 100 50000 100 20998 100 16920

*** Data not available

* (Pham et al., 2006b)

104

3.3.5 Statistical Analysis of the T-Test

An optimisation algorithm is more robust and stable when the variance of a

performance criterion over a number of simulation runs is small (Engelbrecht,

2005). Engelbrecht showed the robustness of a swarm to be in the range:

Robustness(S(t)) = [θ - σθ, θ + σθ]

Equation (2.1)

(Engelbrecht, 2005)

Where θ is the average of the performance criterion over a number of

simulation runs, and σθ is the variance in the performance criterion. The smaller

the value of σθ the smaller the range performance values unto which the

simulations converge – the more stable the swarm.

To check the statistical significance of the result by the Bees Algorithm, a T-

Test is performed. The T-Test checks the relationship between two variables, in

this case two different sets of parameters of the same algorithm and it tries to

answer two questions:

1. What is the probability that a relationship exists?

105

2. If it does, how strong is the relationship?

In other words, tests for statistical significance are employed to address the

question: what is the probability that the relationship between two variables is

really just an occurrence of chance?

T-Tests are often employed in several different types of statistical tests:

- to test whether there are differences between two groups on the same

variable based on the mean (average) value of that variable for each group;

- to test whether a mean (average) value of a group is greater or less than

some standard;

- to test whether the same group has different mean (average) scores on

different variables;

The test is employed for comparing the means of two samples even if they have

different numbers of replicates. The test compares the actual difference between

two means in relation to the variation in the data (expressed as the standard

deviation of the difference between the means). A null hypothesis or an

expectation to test against is required. In this case, for the Bees Algorithm and

the Meta Bees Algorithm, the null hypothesis is that there is no difference in the

performance of the two algorithms. The T-Test will help to decide if the data is

106

consistent with this or departs significantly from this expectation. The T-Test

assesses whether the means of two groups are statistically different from each

other by providing an alpha (α) parameter. The parameter has three ranges:

Where

α < 0.05, there is a significant difference in the group means.

α < 0.01, there is a more significant difference in the group means.

α < 0.001, there is a most significant difference in the group means.

The formula for the T-Test is a ratio. The numerator of the ratio is the

difference between the two means or averages. The denominator is a measure

of the variability or dispersion of the scores.

3.3.5.1 T-Test Results

The first test applied to the obtained results was the T-Test (Schneider and

Kirkpatrick, 2006). The T-Test was applied to all pairs of solutions and the

difference in mean values was found to be significant at the 99% confidence

level. That is, the Meta Bees Algorithm requires significantly fewer evaluations

to solve the benchmarking functions than the Bees Algorithm, as shown in

Figure 3.12 (a-e).

107

To demonstrate this significance graphically, the number of evaluations for

each independent run for both the Bees Algorithm and Meta Bees Algorithm

that were required to optimise the five functions studied in this chapter are

shown in Figure 3.12 (a-e) (in figure (e), the line representing the original

parameters is on the X axis and cannot be shown). It can be seen that on

occasion the Bees Algorithm required fewer evaluations but that on average,

over 100 runs, the Meta Bees Algorithm required fewer evaluations. The T-Test

was conducted on the Meta Bees Algorithm and the original Bees Algorithm.

Table 3.6 shows the alpha values of each test function after applying the T-Test

to these functions.

These values indicate that the results obtained by both the Meta Bees Algorithm

and the original Bees Algorithm are most significantly different with a

confidence level above 99%.

108

Table 3.6. The Values of Alpha of each Test Function

Funct. no. 1 2 3 4 5

Alpha 7.135×10-17

1.148×10-19

2.646×-19

5.735×10-16

1.235×10-15

109

(a) De Jong function

(b) Martin and Gaddy function

(c) Rosenbrock function 2D (a)

0

5000

10000

15000

20000

25000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

solutions

Nu

mb

er

of

Eval

uat

ion

s

Original Parameters

Optimised Parameters

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

solutions

Nu

mb

er

of

Eval

uat

ion

s

Original Parameters


0

10000

20000

30000

40000

50000

60000

70000

80000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

solutions

Nu

mb

er

of

Eval

uat

ion

s

Original Parameters


110

(d) Rosenbrock function 2D (b)

(e) Griewank function

Figure 3.12 (a-e) Student T-Tests for both the Bees Algorithm and Meta Bees

Algorithm

0

50000

100000

150000

200000

250000

300000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

solutions

Nu

mb

er

of

Eval

uat

ion

s

Original Parameters


0

500

1000

1500

2000

2500

3000

3500

4000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

solutions

Nu

mb

er

of

Eval

uat

ion

Original Parameters


111

3.3.5.2 Random Distribution Test

A Random Distribution test was also performed to the both results obtained

using the standard Bees Algorithm and Meta Bees Algorithm to ensure the

quality of the random numbers generated by the system. The Bees Algorithm

includes some randomisation in its two local and global searches such as the

initial distribution of the scout bees over the search space and the recruitment of

bees within the neighbourhood search area. This requires generating a sequence

of random numbers. As the possible amount of generable random number is

finite, after a certain number of calls the sequence of random numbers that has

already been produced is repeated again. Hence, the larger this sequence length

is, the better the number generator should be. Random number generators

always need at least one integer value called the seed to get started. Different

seeds do not usually lead to different sequences of random numbers, but the

random number generator starts at different points in its finite sequence of

random numbers which often leads to different results. Various tests have been

developed to check the quality of randomness of these random number

generators. The Bees Algorithm uses a uniform distribution of the random

number which means that every point of the search space has the same

probability of being chosen and not being biased to certain parts of the search

space otherwise the randomness of the algorithm would be meaningless which

112

will affect the mechanism of how the Bees Algorithm explores new solutions as

discussed in chapter 3.

To test the quality of the random numbers a normalised distribution test was

used (Schneider and Kirkpatrick, 2006). To visualise the distribution in this

test, the interval is divided into certain number of subintervals called bins and

then how many numbers there are in each bin are displayed. Figures 3.13 to

3.17 show two series of histograms that show the distribution of the generated

random numbers during the Bees Algorithm search. One histogram is divided

into 10 bins and the other is divided into 100 bins. For the value of bins equal to

10, some intervals get more random numbers than others, but as bins grow, the

distribution of the random numbers becomes more and more equal among the

intervals and graphs are smoother thus more acceptable randomness in the

generated random numbers is provided for each benchmark.

113

(a) 10 bins

(b) 100 bins

Figure 3.13 Distribution of Random Numbers in both 10 and 100 Bins for the

De Jong Function

0

10

20

30

40

50

60

70

80

90

0 0.2 0.4 0.6 0.8 1

Intervals

Nu

mb

er

of

bin

co

un

t

0

100

200

300

400

500

600

700

800

0 0.2 0.4 0.6 0.8 1

Intervals

Nu

mb

er

of

bin

co

un

t

114

(a) 10 bins

(b) 100 bins


Martin and Gaddy Function

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1

Intervals

Nu

mb

er

of

bin

co

un

t

0

1000

2000

3000

4000

5000

0 0.2 0.4 0.6 0.8 1

Intervals

Nu

mb

er

of

bin

co

un

t

115

(a) 10 bins

(b) 100 bins


Rosenbrock (a) Function

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1

Intervals

Nu

mb

er

of

bin

co

un

t

0

1000

2000

3000

4000

5000

0 0.2 0.4 0.6 0.8 1

Intervals

Nu

mb

er

of

bin

co

un

t

116

(a) 10 bins

(b) 100 bins


Rosenbrock (b) Function

0

100

200

300

400

500

600

700

800

0 0.2 0.4 0.6 0.8 1

Intervals

Nu

mb

er

of

bin

co

un

t

0

1000

2000

3000

4000

5000

6000

7000

8000

0 0.2 0.4 0.6 0.8 1

Intervals

Nu

mb

er

of

bin

co

un

t

117

(a) 10 bins

(a) 100 bins

Figure 3.17 Distribution of Random Numbers in both 10 and 100 Bins for

the Griewangk Function

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1

Intervals

Nu

mb

er

of

bin

co

un

t

0

10

20

30

40

50

60

70

80

90

0 0.2 0.4 0.6 0.8 1

Intervals

Nu

mb

er

of

bin

co

un

t

118

3.4 SUMMARY

The aim of the study reported in this chapter was to characterise the behaviour

of the Meta Bees Algorithm, highlight its strengths and weaknesses and show

the differences between the performance and reliability of the proposed method

and those of competing algorithms.

The search strategy of the Bees Algorithm is to combine global random

exploration with local neighbourhood sampling. Explorative search (using scout

bees) and exploitative search (employing recruited foragers) are clearly

differentiated and they can be independently varied through a set of learning

parameters. This clear decoupling between exploration and exploitation

facilitates the tuning of the Bees Algorithm.

Parameter tuning for metaheuristic search algorithms can be a time-consuming

and inexact way to find appropriate parameter values to suit various classes of

problems. An alternative approach has been explored in this chapter in which

the algorithmic mechanics of the Bees Algorithm are used to produce suitable

values while problems are being solved.

The performance of the Meta Bees Algorithm was evaluated using five

benchmark minimisation tasks. The results were compared with those produced

119

by the Standard Bees Algorithm, the SIMPSA, the NE SIMPSA, the GA and

ACS.

The results for the function optimisation instances used to test this notion

suggest that its performance, in terms of solution costs and run times, is

comparable to the standard implementation in which values from (Pham et al.,

2006b) were employed. In fact the performance in terms of objective cost was

often an improvement over the standard set. This may be attributed to the

ability of the new Meta Bees Algorithm to tailor parameter values to the

problem instance being solved.

The results reveal that the improved solver generally performs well against one

that uses standard/fixed parameter values. This is attributed to the fact that

parameter values suitable for a particular problem instance can be automatically

derived and varied throughout the search process.

The results also highlight the importance for the Bees Algorithm to conduct a

sustained and thorough exploitation of the parameter search space. In addition,

the results show that good quality solutions are achieved for a range of function

optimisation problems.

120

The tests proved the strength of the Bees Algorithm in terms of accuracy,

learning speed and robustness. In four of the five benchmark cases considered,

the Bees Algorithm ranked amongst the top performing optimisation

procedures.

The results reveal that for the function optimisation problem, the use of the

Meta Bees Algorithm solver generally performs well against one that employs a

standard set of parameter values. This is attributed to the fact that parameter

values suitable to a particular problem instance can be automatically derived

and varied throughout the search process.

121

CHAPTER 4

MODIFIED BEES ALGORITHM FOR SOLVING

STOCHASTIC OPTIMISATION PROBLEMS

This chapter describes the application of the Bees Algorithm and statistical

analysis to the solution of a stochastic optimisation. The algorithm employed to

carry out this task was designed with a new fitness evaluation method based on

computing the average fitness value for each bee over a number of trials rather

than computing the value of a single trial. The method enabled the algorithm to

be applied to those situations where the returned value of the fitness function of

a site is different every time this function is called. Hence, there is a need to

evaluate this site a number of times by sending a number of bees to the same

site (point) and then calculating their average value. This means that the

number of bees sent to each site is not constant and it is based on feedback

information gathered from the bees recruited so-far to the same point. To test

the algorithm, the parameter value optimisation of a metaheuristic method is

employed. This test took the shape of parameter setting for the ACS algorithm

which is employed to solve a TSP as a stochastic problem.

The chapter is organised as follows: the new modified Bees Algorithm for

solving stochastic optimisation problems is presented in the next section.

Section three details the analysis of the algorithm and the steps that it takes to

122

converge to the near optimum solution. The application of the algorithm to

parameter setting for the ACS-TSP problem and a description of the

experimental conditions concludes the chapter.

4.1 PROPOSED MODIFIED BEES ALGORITHM FOR SOLVING

STOCHASTIC PROBLEMS

This section proposes a modified Bees Algorithm for solving stochastic

problems. The algorithm works like the original Bees Algorithm (see Figure

3.1) but with the inclusion of a new statistical part which leads the Bees

Algorithm to evaluate the performance of each bee (see Figure 4.1). The

original Bees Algorithm involves calling the fitness function of the site where

the bee has been placed either randomly or directly within the patch. In

deterministic problems such as those the Bees Algorithm has dealt with before,

the value of the fitness function is fixed for the same site visited by a bee and

hence evaluating it only once is enough. However, with the new algorithm, the

modified Bees Algorithm is tested on stochastic problems where the fitness

function consists of a set of parameters which change randomly. This means

that every time the fitness function is called, its return value will be different so

there is a need to evaluate each site a number of times by sending a number of

bees to the same site and then calculating their average value. The number of

bees sent to each site is not constant and it is based on feedback information

gathered from the bees recruited so-far to the same point. The feedback

123

information that will determine the number of bees sent to the same point is

based on the slope of the line formed from the linear regression of the fitness

values of the sent bees. The algorithm will stop sending more bees to the same

site when it indicates a near horizontal line. By adopting this technique, a more

accurate average fitness value for the sent bees to the same site is guaranteed

while at the same time consuming the minimum number of evaluations. The

accuracy of this average value will depend on the minimum number of bees

sent to the same point and the minimum acceptable slope value.

Figure 4.1 shows the pseudocode of the new algorithm. It is exactly like the

standard Bees Algorithm, the only difference being in the way that a bee is

evaluated. A subroutine replaces the direct evaluation method. The acceptable

slope value will determine the accuracy and the speed of convergence of the

algorithm. If the acceptable slope value is very small, the algorithm will take a

longer time to converge but its accuracy will be very high. On the other hand, if

a higher acceptable slope value is chosen, the running time will drop, but so

will the accuracy.

124

1. Initialise population with random solutions.

2. Evaluate fitness of the population by calling the subroutine.

3. While (stopping criterion not met)

// forming new population.

4. Select sites for neighbourhood search.

5. Recruit bees for selected sites (more bees for best e sites) and evaluate

fitness by calling the subroutine.

6. Select the fittest bee from each patch.

7. Assign remaining bees to search randomly and evaluate their fitness.

8. End While.

// Subroutine for bees fitness evaluation.

1. Evaluate fitness.

2. While (minimum slope value not reached)

3. Evaluate the fitness and store its value.

4. End while.

5. Calculate the mean from the stored values and return it.

6. End Subroutine.

Figure 4.1 Pseudocode of the Modified Bees Algorithm for Solving Stochastic

Problems

125

4.2 COMPUTATION EXPERIMENTS

The experimental analysis was conducted in three stages; the first stage was to

answer the question of why there was a need to evaluate the same point in a

search space more than once in a stochastic problem. To address this, a study

on the parameters of the ACS and their influence on the speed of convergence

of the TSP instance was performed. Secondly, a study and analysis was

conducted on when to stop the evaluation of a point. This was achieved by

studying the relationship between the slope of the line formed from the linear

regression of the average fitness values of the sent bees and the accuracy of

these values obtained by the algorithm. Finally, based on these two previous

stages, the proposed method was applied on the ACS to find the best

combination of parameters that guarantees the fastest convergence to the best

solution on a TSP instant. The stages outlined above are now discussed in

detail.

4.2.1 Analysis of the Speed of Convergence of ACS

The speed of convergence is represented graphically in the following figures for

two different experiments with different sets of parameter values. Figure 4.2

shows the first experiment on the variation of the number of function evolutions

that the ACS uses to find the optimum solution of a TSP instance called

126

Oliver30 (Oliver et al., 1987) and is illustrated in Figure 4.3 over 50 runs using

the same set of parameters shown in Table 4.1. Statistical information, see

Table 4.2, shows that the average value of the fitness which represents the value

of speed of convergence is 463.

Figure 4.4 shows the second experiment using a different set of parameters as

shown in Table 4.3 for the same TSP instance. Table 4.4 shows statistical

information from this experiment where the average value over 50 runs is 2370.

127

Table 4.1 Parameters employed in the First Experiment

m β ρ q0

11 4 0.2 0.7

128

Figure 4.2 Fitness Values for the First Experiment over 50 Runs

-200

0

200

400

600

800

1000

1200

1400

1600

1800

0 10 20 30 40 50 60

Runs

Fitn

ess

129

Figure 4.3 Oliver30 TSP Problem

130

Table 4.2 Statistical information for the first experiment

Runs Min Max Average

50 22 1584 463

131

Table 4.3 Parameters employed in the Second Experiment

m β ρ q0

25 13 0.7 0.2

132

Figure 4.4 Fitness Values for the Second Experiment over 50 Runs.

-2000

0

2000

4000

6000

8000

10000

12000

14000

0 10 20 30 40 50 60

Runs

Fitn

ess

133

Table 4.4 Statistical Results from the Second Experiment

Runs MIN MAX Average

50 275 13050 2370

134

The above two experiments demonstrated:

1. The importance of tuning the parameters of the algorithm to generate a

higher quality solution (fitness function) because for every set of

parameters totally different statistical values (minimum, maximum and

average) are produced, and

2. Although the same set of parameters were employed for each of the two

experiments over 50 runs each, the fitness value was never the same.

This indicates, unlike with deterministic problems, the importance of

evaluating the same point (a set of parameters in this case) several times

and hence the need for the proposed method. In summary, this revealed

that the parameter setting for the metaheuristic method can be

considered a stochastic optimisation problem and it is affected directly

by its parameters values.

4.2.2 Analysis of Calculating the Average Fitness

To demonstrate the relationship between the slope of the line formed from the

linear regression of the average fitness values of the sent bees and the values

obtained by the algorithm, an analysis was performed on a stochastic

135

optimisation problem. Figures 4.5, 4.6 and 4.7 show three different variations

of fitness values obtained from many evaluations of the same stochastic

problem. Figure 4.5 shows the variation of fitness values for five runs. In this

figure the slope is not equal to zero, which means more evaluations and fitness

values are needed to obtain a more accurate average value. This means that five

evaluations are not sufficient to calculate the precise average value.

Figure 4.6 shows the variation of fitness values obtained from evaluating the

same stochastic problem over 22 runs, the slope of the line formed is near to

zero and the line is nearly horizontal. When the average value of slope of the

line for 22 runs is compared to the average of 90 runs, Figure 4.7, it shows that

they are nearly the same.

The slope of the line formed by applying a least-squares liner regression on the

bees sent to the same point is an indication of the number of evaluations that is

required of this particular point to be able to represent a more accurate average

value of its fitness. By using this type of feedback in the modified Bees

Algorithm, it will ensure having a more accurate average value while requiring

only a minimum number of evaluations. In conclusion, adding the statistical

feature to the Bees Algorithm allows it to solve stochastic optimisation

problems.

136

Table 4.5 shows the average obtained for three different numbers of runs; 5, 22

and 90 runs, where the average values found were 658, 472 and 471

respectively.

137

Figure 4.5 Non-Zero Slope of Regression Line Shows an Insufficient Number

of Evaluations for Calculating the Average Value

y = 92.4x + 380.6

R2 = 0.0881

0

200

400

600

800

1000

1200

1400

0 1 2 3 4 5 6

Runs

Fitn

ess

138

Figure 4.6 Number of Evaluations - a Representation for 22 Runs

y = 0.4457x + 493.48

R2 = 5E-05

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20 25

Runs

Fitn

ess

139

Figure 4.7 Number of Evaluations - a Representation for 90 Runs

y = -0.0953x + 476.6

R2 = 3E-050

500

1000

1500

2000

2500

3000

3500

0 20 40 60 80 100

Axis Title

Axi

s Ti

tle

Fit

ness

Runs

140

Table 4.5 Fitness Values for 5, 22 and 90 Runs

Number of Runs

Fitness Value

MAX MIN Average

5 1232 88 658

22 1584 22 472

90 3267 22 471

141

4.2.3 The Proposed Bees Algorithm

The problem of setting parameters can now be solved using the modified Bees

Algorithm. To examine the efficiency of the modified Bees Algorithm, it is

tested on the problem of finding the best parameter values for the ACS by

solving TSP instances.

The fitness value is the total number of evaluations consumed by the ACS to

find the optimum solution of the TSP instance (see Figure 4.8).

The ACS parameters chosen to be optimised are the number of ants (m), the

parameter that determines the relative influence of the heuristic value (β), the

parameter that balances between exploration and exploitation behaviours of the

algorithm (q0) and the parameter that governs the local pheromone evaporation

(φ). The search spaces for these parameters are shown in Table 4.6. The other

parameters (the length of the candidate list (cl = 10), the parameter that governs

global pheromone decay (φ=0.1) and the initial value of the pheromone trails

value (τ0) are initialised by their default values as mentioned in section 2.4.6.3.

142

Figure 4.8 Layers Showing the Bees Algorithm Wrapping the ACS-TSP

143

Table 4.7 shows the best parameters obtained by applying the modified Bees

Algorithm on the ACS-TSP Oliver30 dataset. A comparison of the results with

other research found in the literature is also presented.

Experiments were produced on a 2.00GHz Pentium Dual Core processor with

1GB of RAM. The process took three days to arrive at the best parameters for

the Oliver30 dataset using the programming language C#.Net 2008 and the .Net

framework 3.5.

144

Table 4.6 Search Spaces of the ACS Parameters

Parameter m β ρ q0

Min value 2 0 0 0

Max value Number of Cities 14 1 1

145

Table 4.7 The Optimum Results Obtained from the Modified Bees Algorithm Compared to

Suggested Values in the Literature

References m β ρ q0 Mean no.

of evaluations

(Dorigo and Gambardella, 1997) 10 2 0.1 0.9 7839

Pilat and White (2002) 20 6 0.2 0.7 7443

Modified Bees Algorithm 12 4 0.2 0.8 2984

146

4.3 SUMMARY

The chapter has described a method to solve stochastic optimisation problems.

The key operation in this method was the multiple evaluation of the fitness of

each parameter setting tried, when using a modified version of the Bees

Algorithm to optimise stochastic problems based on statistical analysis. The

new algorithm was tested on the problem of finding the best values of the

parameters for an ACS-TSP.

The results obtained show that the algorithm was able to effectively find near

optimum solutions of such stochastic optimisation problems. Also, the results

show that parameter setting produced by the Bees Algorithm outperformed

parameter sets that were suggested by other parameter settings methods such as

the ACS and the GA

Further work could investigate the possibility of using the variation in the

values of the fitness with a particular parameter setting as an indicator of the

stability of that setting. For such an investigation, it might be useful to adopt a

multi-objective optimisation technique such as that presented in

(Ghanbarzadeh, 2007). Also, this modified Bees Algorithm can be applied to

many well-kown TSP benchmarking instances for finding the best parameters

147

for them by taking the average of these optimum parameters for different

instances.

148

CHAPTER 5

APPLICATION OF THE BEES ALGORITHM TO THE

SOLUTION OF COMBINATORIAL OPTIMISATION

PROBLEMS

This chapter discusses the use of the Bees Algorithm to solve combinatorial

optimisation problems. The problems belong to the general class of Travelling

Salesman Problems. In particular, the application studied is that of optimising

the scheduling of operations for a printed circuit board (PCB) assembly

machine.

5.1 BEES ALGORITHM CHARACTERISTICS FOR

COMBINATORIAL OPTIMISATION PROBLEMS

In general, the Bees Algorithm is a population-based stochastic search

algorithm designed to solve specific types of optimisation problems

(Ghanbarzadeh, 2007). When the Bees Algorithm is applied to combinatorial

optimisation problems, these problems are generally characterised by the

following:

the search space is discrete;

there is a set of finite constraints;

a solution is represented as an ordered sequence of components;

149

there is a cost function which associates a cost to each solution generated by

the search algorithm, where each component added to a solution contributes to

the total cost of the solution;

there is a finite set of components from which solutions are constructed;

there is a finite set of possible transitions among the complements;

there is a finite set of sequences of components representing all possible

valid combinations of components.

Given these characteristics, the purpose of the Bees Algorithm is to construct a

feasible sequence of components such that the cost of the solution is minimised

when implemented in the minimisation problem (Pham et al., 2007a) and such

that the sequence of components represents only valid transitions.

The Bees Algorithm applied to combinatorial problems requires a graphical

representation of these problems i.e. a graph that consists of a finite number of

nodes and links between nodes (Engelbrecht, 2005). Each node represents one

of the components, and a link represents a transition from one node to the next.

A cost is associated with each link. The objective of the Bees Algorithm is to

traverse this graph, in order to construct a minimum cost path. The constructed

150

path represents a sequence of components, i.e. a solution to the optimisation

problem. In the case of the Bees Algorithm, a path is constructed by randomly

generating all the nodes at once.

The Bees Algorithm models a colony of honeybees searching for multiple

solutions in parallel. To achieve the task of constructing optimal paths, bees

have the following properties and characteristics:

A bee has a memory to store information on the path constructed. The memory

is employed mainly to enforce constraints, such as the inclusion of a component

only once.

One or more termination conditions are associated with each bee. These

conditions include: ‘A solution with acceptable cost has been constructed’, ‘A

maximum number of iterations has been exceeded’ or ‘Stagnation behaviour is

observed’.

To demonstrate the characteristics of the proposed algorithm for solving

combinatorial problems, the Bees Algorithm has been applied to the TSP as it is

easy to understand, has numerous applications and has been studied extensively

by researchers from various disciplines (Aarts et al., 1988; Chandra et al., 1999;

Freisleben and Merz, 1996; Gambardella and Dorigo, 1996; Helsgaun, 2000;

151

Knox, 1994; Laporte, 1992). In addition, the TSP is considered difficult to

solve as it is a non-deterministic polynomial-time (NP-hard) problem, which

means that the time complexity of finding an optimal solution to these types of

problems grows exponentially with the problem size (e.g. the number of cities)

(Laporte 1992). In a TSP with N cities, there are

2

! 1N possible tours that

must be computed in order to determine the optimal path. Rather than searching

all possible tours to find an optimal solution, a common approach for this class

of problem is to find a solution that is “good enough”.

Many of these “satisfying” solutions can be determined in polynomial time.

The TSP is a problem of finding the shortest tour that visits all the nodes of a

fully connected graph, the nodes of which represent locations, and the arcs

representing paths with associated costs (normally assumed to be distance).

5.2 THE BEES ALGORITHM FOR SOLVING THE TRAVELLING

SALESMAN PROBLEM

The TSP has been one of the most popular combinatorial optimisation problems

studied. A TSP can be either symmetric or asymmetric. A TSP is considered

symmetric if travelling from city A to city B is given the same weight as

travelling from city B to city A. This would be the case if we only consider

distance, however if other factors are taken into account then we may not be

152

able to assign the same weight to travelling in both directions between two

cities. Such cases are called asymmetric TSPs or ATSPs (Freisleben and Merz,

1996). There are numerous techniques employed for tackling this class of

combinatorial optimisation problem, including methods that are specifically

tailored to the TSP as well as more general-purpose metaheuristics which can

be applied to a number of hard combinatorial problems such as the graph

partitioning problem and the quadratic assignment problem (Garey and

Johnson, 1979).

In the subsequent section, the main steps required to implement the Bees

Algorithm when applied to solve the TSP will be presented.

5.2.1 Implementing the Bees Algorithm for the Travelling Salesman

Problem

The TSP can be represented as a sequence of N cities to be visited, where the

actual order of the sequence determines a particular solution to the problem

(Johnson and McGeoch, 1997). Thus in general the search space consists of all

N! permutations.

The Bees Algorithm can be applied to the TSP in a straightforward way. Where

each bee represents a candidate tour and where the first and the last element of

153

the bee represent the city of origin. For example, the bee b (B, D, A, E, C, B)

represents a tour from city B, via cities D, A, E and C, then back to city B.

When applying the Bees Algorithm to a TSP as shown in Figure 5.1, the

algorithm starts with an initial population of n scout bees randomly distributed

in the search space by generating a set of sequences representing the visiting

sequence of each bee. Each bee becomes a symbolic string representing the

sequence of cities. For N cities to be visited by a salesman, a string with a

length of N is needed to encode each candidate solution where each bee will

visit (C) number of cities generated randomly from one to (C), which

formulates (C) links employed as the initial population for each bee.

To generate b(B, D, A, E, C, B), a random number generator is employed to

generate letters between A and E and once a letter is generated (i.e. B in this

example) it will not be allowed to appear again. The first letter generated will

appear again at the end to give a closed tour. Then the generator generates

another letter (D in the above example), and so on until all the five letters are

generated to constructed a full and closed tour.

In step 2, the fitness computation process (i.e. the performance evaluation of the

candidate solutions) is carried out for each patch visited by each bee. This is

completed by calculating the distance between each two adjacent cities in the

154

sequence and adding all the distances together to find the total length of each

tour using the objective function shown in Equation 5.1:

),(),()( )1()(

1

1

)1()( ccdccdf N

N

i

ii

Equation 5.1

Where d(ci,cj ) is the distance between cities i and j, and (i) for i=[1], N

defines a permutation.

For the next step, step 3, the m patches (i.e. the tours) with the highest fitness

levels are designated as “selected patches” and chosen for a neighbourhood

search.

Once completed, step 4, is commenced. The algorithm searches around the

selected patches using a local search algorithm. Various operators could be

employed to create neighbours to a given bee, including monadic operators

such as mutation, inversion, swap and insertion (single or multiple). For the test

problems considered in the next section, the single-point insertion and the two-

paths optimal (2-Opt) operators (Burke et al., 1999) are adopted and shown.

In step 5, each tour created by a local operator represents a neighbour to the

selected patches, and more tours will be created for the best e patches. In step 6,

only the bee with the highest fitness (lowest tour length) in each patch will be

155

selected to form the next bee population. In step 7, the remaining bees in the

population are placed randomly around the search space to scout for new

potential solutions (tours).

Steps 4-7 are repeated until either the best fitness value (which will be

designated as the ‘global best tour’) has been found or the specified maximum

number of iterations has been reached.

At the end of each iteration, the colony will have two parts to its new

population: representatives from the selected patches, and scout bees assigned

to conduct random searches. These steps are repeated until a stopping criterion

is met.

156

1- Initialise population with random solutions.

2- Evaluate fitness of the population.

3- While (stopping criterion not met)

//Form new population.

4- Select sites for a neighbourhood search using a neighbourhood operator.

5- Recruit bees for selected sites (more bees for best sites) and evaluate fitness.

6- Select the fittest bee from each patch.

7- Assign remaining bees to search randomly and evaluate their fitness.

8- End While loop.

Figure 5.1 Pseudocode of the Standard Bees Algorithm

157

5.2.2 The Bees Algorithm with Neighbourhood operators

Given a feasible solution, x, to the problem, its neighbourhood is a set of

solutions obtained from x by disturbing its components in some specified

manner (Ibaraki, 1997). Two neighbourhood operators for the TSP have been

implemented for the Bees Algorithm. These are:

A. Single-point insertion

Illustrating this process with an example, a closed sequence C B E A D C

shown in Figure 5.2a, will be modified in the following way. The insertion

operator breaks up the sequence by removing a section, in this case BE,

between two randomly chosen positions B and E. This section is then inserted

at a randomly selected point within the sequence; in this case, it is placed after

point D. It must be noted that this is carried out while preserving the visiting

order of the positions on the inserted section, producing the new sequence: C A

D B E C. The sequence is then reconnected, as shown in Figure 5.2b.

158

Original closed sequence : C B E A D (C)

Sequence after removal of section (BE) : C A D (C)

Sequence following insertion of (BE) after point D : C A D B E (C)

Figure 5.2a Single-Point Insertion

Figure 5.2b Single-Point Insertion

A

D

E

B C

A

D

E

C B

A

E

C B

D

159

B. The 2-Opt operator

The two (two paths) optimal (2-Opt) algorithm is a tour improvement procedure

(Okano et al., 1999). The best-known tour improvement procedures have been

found to be edge exchange procedures (Lin and Kernighan, 1973) such as the k-

Opt algorithm where all exchanges of k edges are tested until there is no

feasible exchange that improves the current solution; this solution is said to be

k-optimal. Since the number of operations increases rapidly with increases in k,

k = 2 and k = 3 are most commonly employed.

A 2-Opt operator involves randomly breaking a sequence into two paths. The

visiting order of the positions (cities) on one of the paths is then reversed before

the two paths are reconnected. The application of a 2-Opt operator to a closed

sequence of five positions (cities) can be seen in Figure 5.3a below.

In addition to the use of single-point insertion, the 2-Opt local search algorithm

will be applied to the Bees Algorithm for the TSP. In the 2-Opt algorithm, two

tours are considered neighbours if one can be obtained from the other by

deleting two edges, reversing one of the resulting two paths, and reconnecting

them. In the TSP, the 2-Opt neighbours of the tour are pair-wise part exchanges

within the current solution sequence. For example, with a number of cities n = 5

160

and an original solution sequence of (C, B, D, A, E) shown in Figure 5.3b, it

will be modified in the following way. Two edges will be deleted randomly

(e.g. BD and EC), thus breaking the sequence into two paths (CB) and (DAE).

One of the paths (DAE) will be reversed to become (EAD) and then the two

paths will be reconnected. The new sequence is C B E A D. The sequence is

then reconnected.

161

Original closed sequence: C B D A E (C)

Broken-up sequence: C B -- D A E

Path 1: C B

Path 2: D A E

Path 2 (after reordering): E A D

Re-connected sequence: C B E A D (C)

Figure 5.3a 2-Opt Operator

162

Figure 5.3b 2-Opt Operator

A

E

D

B

C

B

C

A

E

D

A

E

D

C B

163

In addition to the random way, there are different ways to generate a starting

tour for the initial application of the 2-Opt algorithm. A typically way is to use

a random number generator. In addition, a greedy solution obtained by a greedy

algorithm can be employed. For example, applying a greedy algorithm (such as

the Nearest Neighbourhood algorithm) to the TSP yields the following

instruction: "At each stage visit the unvisited city nearest to the current city"

(Lawler et al., 1985). In this study however, the starting point is rapidly attained

using the Bees Algorithm as explained in Section 5.2.1.

5.2.3 Experiment Results using the Bees Algorithm

For the purpose of comparison with other optimisation techniques, the Bees

Algorithm was tested on three specific benchmark TSPs. The dimensions of

these problems were 16, 44 and 91 cities with optimum solutions equal to

377.006mm, 569.706mm and 581.421mm respectively. The Euclidean metric

which is the distance between two points is employed in these examples. Table

5.1 presents the results produced by the Bees Algorithm along with those

produced by three other approaches for the same TSPs. These approaches are

the GA, SA and Ant Colony System (ACS) (Stützle and Hoos, 1997).

164

Table 5.1 Performance of the Bees Algorithm Compared to the GA, SA and ACS

The algorithm

TSP instances

16 Cities 44 Cities 91 Cities

No. of

iterations

Tour length No. of

iterations

Tour length No. of

iterations

Tour length

GA 269 377.006 1908 593.689 4780 721.074

SA 4810 377.006 97392 670.938 307921 884.009

ACS 213 377.006 9 569.706 7 581.421

Bees Algorithm 190 377.006 450 569.706 500 669.096

165

The results show that the Bees Algorithm is able to find the optimal solution for

two of the benchmark problems (16 cities and 44 cities) and a near optimal

solution to the third problem. The Bees Algorithm required less time (a fewer

number of iterations) to find the solution to all three problems than either the

GA or SA. However, while it could outperform the ACS for the smallest sized

problem (16 cities), it needed more time for larger numbers of cities. Table 5.1

also shows that the performance of the Bees Algorithm generally drops when

the size of the problem instances increases. One of the reasons could be the

parameter settings that need to be fine-tuned to cater for different scenarios.

The next part of this chapter discusses the use of the Bees Algorithm to solve

more complex optimisation problems than the TSP and that is the Printed

Circuit Board (PCB) assembly planning problem. This is carried out over four

sections:

The first part, Section 5.4.1, reviews the basic types of PCB assembly machines

available and the assembly planning problems to which optimisation techniques

have been applied. The second part, Section 5.4.2, describes the general use of

the Bees Algorithm to generate optimal PCB assembly plans. The third part,

Section 5.4.3, demonstrates the application of this algorithm to one type of

placement machine. The final part, Section 5.4.4, presents the results obtained.

166

5.3 THE BEES ALGORITHM AS APPLIED TO THE PRINTED

CIRCUIT BOARD ASSEMBLY OPTIMISATION PROBLEM

PCB assembly is the process of placing electronic components of different

shapes and sizes employing various types of surface mount technology (SMT)

placement machines at specific locations on a PCB. These placement machines

have the ability of fast component placement and can handle high and rapid

production demands. In its simplest form, PCB assembly optimisation aims to

minimise the time needed to process the different components. Similar to the

TSP, this time grows exponentially with the problem size (i.e. the number of

components).

In order to make full use of the high speeds of these placement machines, many

intelligent optimisation techniques have been applied to find the best-solution

to the PCB assembly problem by optimising their operation. These optimisation

techniques include GAs (Goldberg, 1989), SA (Mathur et al., 2000), and

Evolutionary Programming (EP) (Nelson and Wille 1995). Results have shown

that these techniques are able to provide near-optimal solutions in a short period

of time.

167

5.3.1 Printed Circuit Board Assembly

A. A PCB Assembly Machine

A PCB assembly machine in its generic representation comprises of three parts:

- a feeder F1 (or an array of feeders F2 ) which supplies components,

- an assembly head H which picks up components from F1 or F2 and

places them onto the PCB,

- a table T which carries the PCB.

Table 5.2 illustrates three types of machines for placing through-hole and

surface-mount components in sequence onto a PCB (Ayob et at., 2002).

In a type–1 machine, a single feeder F1, which can take the form of a magazine,

provides components to a single assembly head H (there is no turret) at a fixed

location in the horizontal x-y plane. The head H also deposits components onto

the PCB at a fixed point in the x-y plane. The table T moves in the x-y plane so

that components are placed at the desired locations on the PCB. This type of

assembly machine is the simplest and is normally adopted when only one type

of component needs to be placed.

168

In a type–2 machine, the feeder (or feeder array) and the table are stationary

(and hence the PCB is also held in at a fixed location). The single assembly

head H (again, there is no turret) moves in the x-y plane carrying components to

the correct placement positions on the PCB. If an array of feeders is adopted,

the machine can be employed to place components of different types onto a

PCB.

A type–3 machine is the most complex as it has three moving parts:

(1) A turret carrying multiple pick-up/placement assembly heads H. The centre

of rotation of the turret is fixed in the x-y plane.

(2) A feeder array F2 that moves along the x-axis to bring the feeder with the

required component to the fixed pick-up location.

(3) A table T that moves in the x-y plane to position different points of the

PCB, in sequence, at the fixed component placement location.

Component pick-up and placement occur simultaneously after the correct

feeder and the table have reached their designated positions and the turret has

completed indexing the appropriate pick-up and placement heads (Ayob et al.,

2002). This type of machine can also place components of different types. The

169

synchronisation of the multiple feeders, the multi-head turret pick-and-place

system and the assembly table is required to perform the task.

170

Table 5.2 Types of PCB Assembly Machines and their Characteristics (Ang et al., 2009), T:

Table, F1: Single Feeder, F2: Multiple Feeders, H: Head. Machine Type Type–1 Type–2 Type–3

Model of

system

Model of

problem

Travelling salesman

Pick and place Moving Board with Time Delay (MBTD)

Description of

assembling

process

The machine assembles the components while only the table

T that holds the PCB may move

in the x-y axis. The component is fed directly into the assembly

head H.

The head H moves in the x-y axis to pick and place (assemble) the components

while the table T that holds the PCB is

stationary. The feeder array F2 is also stationary.

The multi-head turret H picks the components from the feeder array F2 (which moves in a

single axis to provide the right component for

assembly) with one head while placing them with the other onto the PC board after rotating.

The PCB is held by a table that moves in the x-y

axis in accordance with the sequence and location of the component that needs to be

assembled.

Key

characteristics

of solution

1. The path to assemble the components

1. The feeder slot arrangement. 2. The shortest path to assemble the

components.

1. The feeder slot arrangement. 2. The shortest path to assemble the

components.

3. The number of heads on the assembly turret.

H

F1

T F2

Turret

F2

171

B. Printed Circuit Board Assembly Planning

PCB assembly planning involves two types of tasks: set-up management and

process optimisation. In the context of planning for a single assembly machine

with an array of feeders (type–1 machine), set-up management can include

arranging the allocation of components among the different feeders which is

referred to as the component allocation problem (CAP). In an alternative

arrangement, the relative positioning of the feeders in the array requires setting

up, this is referred to as the feeder arrangement problem (FAP). Both

arrangements are set-up in a manner to reduce assembly cycle times.

Process optimisation for an assembly machine is usually a component

sequencing problem (CSP). The aim of component placement sequencing is to

optimise the movements of the table (in a type–1 machine) or the assembly

head (in a type–2 machine). This problem could be viewed as a typical TSP for

which the objective function might be expressed as in Equation 5.1a:

N

i

iiTotal CCDD1

1),(

Equation 5.1a

(Ho and Ji, 2007b)

172

Where N is the total number of components to be placed onto a PCB and D(Ci,

Ci+1) is the distance travelled by the table or the assembly head when moving

from the placement position for component Ci to the placement position for

component Ci+1. (Note that the placement position for CN+1 is usually taken to

be the same as that for C1, the starting position, as CN is the last component to

be placed onto a given PCB.)

As the cycle time is the real factor of interest in this problem, the objective

function is usually rewritten as in Equation 5.1b:

V

DT Total

Total

Equation 5.1b

(Ho and Ji, 2007b)

Where V is the average speed of movement of the table or assembly head.

In a type–3 machine, known as the Moving Board with a Time Delay (MBTD),

as shown in Figure 5.4, the assembly time is affected by three factors. These are

the movement of the PCB, the shifting time of the turret head, and the travelling

time of the feeder carrier. The total assembly time needed for a PCB is the

summation of the dominating times associated with these three factors for all

board components. The formula for the total assembly time is defined in

Equation 5.2.

173

N

i

iTotal TT1

Equation 5.2

(Ho and Ji, 2007b)

Where N is the total number of components to be placed and Ti, the time

required to place the component Ci, is given by Equation 5.3.

]),,(),,(max[ 31211 tfftcctT gigiiii Equation 5.3

(Ho and Ji, 2007b)

Where g is the number of assembly heads positioned between the pick-up and

placement heads on the turret, t1 is the time for the table to move from the

location of component Ci-1 to the location of component Ci. The location of C0

is the starting position of the table, and is given by the Chebyshev metric:

y

ii

x

ii

iiv

yy

v

xxcct

11

11 ,max),(

Equation 5.4

(Ho and Ji, 2007b)

Where xi = x coordinate of the component ci, and yi = y coordinate of the

component ci,, vx and vy are the velocities of the x-y table in the x and y

directions respectively (both velocities are assigned as 60mm/s for this study),

174

t2 is the time for the feeder array to change the pick-up feeder from the feeder

supplying component Ci+g to that supplying component Ci+g+1.

Ci+g+1 is the component that is to be picked-up when component Ci is placed

onto the PCB. (As CN is the last component to be assembled onto a given PCB,

CN+g and CN+g+1 are the gth

and (g+1)th

components to be placed onto the next

PCB.) The Chebyshev metric essentially takes into consideration the x-y

movement of the board table as independent motions. This is usually the case

when the x-y table is controlled by two motors, each separately controlling the

x and y movement.

The distance between feeders is measured using the Euclidean metric. The

travelling time of the feeder carrier between feeder ),( f

i

f

ii yxf and ),( f

j

f

jj yxf is

given by Equation 5.5.

f

f

i

f

j

f

i

f

j

jiv

yyxxfft

22

2 ),(

Equation 5.5

(Ho and Ji, 2007b)

Where xi is the x coordinate of the feeder fi, yi is the y coordinate of the feeder

fi, and vf is the speed of the feeder carrier (vf is 60mm/s in the case study).

175

For the case study in this chapter, the feeders are arranged in a straight line (y-

axis) as shown in Figure 5.5 and separated by a y-distance of 15 mm. Hence,

the x-coordinates for the feeders are the same throughout, and any variation due

to any possible vibration will be negligible.

t3 is the time taken by the turret to index the assembly heads by one position

and is set as 0.25 seconds per step. Generally, indexing takes place one position

at a time and always in the same direction.

As there are three moving parts in this type of machine (a board, a feeder, and a

turret), each moving part has to wait for the other two parts to complete their

movements before the next component can be picked up or placed. Hence, the

time Ti, defined in Equation 5.3, needed for the placement of component i is the

maximum time between the board movement t1, the feeder movement t2, and

the indexing time during each pickup and placement t3. The problem with this

type of machine is that it needs to determine simultaneously the placement

sequence of the board components and assign the various types of components

to the feeders such that the total assembly time is minimised (Ho and Ji, 2007a).

Another aspect of this process that must be addressed is the more complex, but

also more realistic, problem of optimising both the arrangement of feeders

denoted as, F = {f1…, fj, …, fR-1, fR}, where fj is the feeder for the jth

component

176

type, and the sequence of placement of components denoted as C ={c1…, ci,

…, cN-1, cN}, where ci is the ith

component to be placed (Figure 5.5) in order to

minimise the total assembly time TTotal as given by Equation 5.2.

177

Figure 5.4 PCB Assembly Machine of the MBTD Type (with 2 Rotary Turret Heads, 10 Feeder Slots and a

Moveable Assembly Table)

Multi-head

turret (with 2

heads in this

diagram)

The y-axis

movement of the

feeder

The x-axis movement of the PCB mounting

table

Feeder

system

with 10

slots

PCB

PCB mounting

table

The y-axis

movement of

the PCB

mounting

table

The rotary movement

of the multi-head turret

178

Component Assembly Link Feeder Arrangement Link

c1 c2 c3 c4 . . . cN-3 cN-2 cN-1 cN f1 f2 . . . fR-1 fR

Figure 5.5 Representation of a PCB Assembly Sequence

179

Previous researchers have applied a variety of techniques to provide a solution

to this PCB assembly planning problem. The techniques employed have

included the GA (Maimon and Brha, 1998; Wong and Leu, 1993; Ong and

Khoo, 1999; Ho and Ji, 2007b), EP (Nelson and Wille, 1995), the minimal

spanning tree technique (Leipala and Nevalainen, 1989), and rule-based expert

system techniques (Yeo et al., 1996). The GA has been successfully employed

to solve the Moving Board with a Time Delay (MBTD) problem (Wong and

Leu, 1993; Ong and Khoo, 1999). Four genetic operators and two-links were

employed to provide solutions for the component placement and feeder

assignment problems. It was shown that the method is easily adaptable to the

planning problems of many types of assembly machines. In (Ong and Tan,

2002), the researchers focused on the application of the GA to solve the MBTD

problem for a high-speed PCB assembly machine where eight operators (four

crossover operators and four mutation operators) were employed. In another

paper (Yeo et al., 1996), a rule-based frame system for PCB assembly was

developed to generate the component feeder arrangement and placement

sequence for concurrent chip placement machines. The system was

implemented using an AI programming environment and led to significant time

savings in PCB assembly. Ho and Ji (2007b) in their book, focused on the

optimisation of the PCB assembly line efficiency. They integrated the

180

component sequencing and the feeder arrangement problems together for the

chip shooter machine.

Of these four techniques, the GA and EP seem to be the most popular due to

their simple implementation and robustness against local optimum traps where

a local, but not global minimum is found. A comparison of the results obtained

by other researchers using the GA and EP on a benchmark PCB assembly

planning problem will be presented. These results were compared with the

results found when applying the Bees Algorithm to the same problem.

5.3.2 The Proposed Bees Algorithm for PCB Benchmark Problems

Figure 5.6 shows the flowchart of the Bees Algorithm when applied to the

Component Sequencing Problem (Type–3 PCB machine). After the parameters

of the Bees Algorithm have been initialised; the algorithm generates initial

solutions (i.e. scout bees) for both component sequencing and feeder

arrangement problems. Hence each bee is comprised of two links (as shown in

Figure 5.5). The first link represents the sequence of component placements and

is generated as a simple TSP. The second link representing the feeder

arrangement is generated randomly.

181

The scout bees are normally created by randomly generating feeder

arrangements and placement sequences and then checking that they are valid. A

valid feeder arrangement would be a permutation of feeder labels, each

corresponding to a particular feeder and component type. For example, label

(A) might correspond to feeder (A) which supplies 100 resistors. The

number of labels in a valid feeder arrangement would be equal to the number of

feeders in the assembly machine. A valid placement sequence would also be a

permutation of labels. Each label representing a placement position on the PCB,

with the total number of labels (i.e. the length of the sequence) being equal to

the number of placement positions.

Next, the fitness of each bee is measured, where each fitness is given as the

assembly time, found using Equation 5.3. The fitness values are then sorted and

the bees with higher fitness values are selected and a neighbourhood search is

initiated using the 2-Opt local search heuristic for the first link and the single-

point insertion algorithm for the second link. The fittest bee from each site is

then selected to form the next bee population. The remaining bees in the

population are placed randomly around the search space to scout for new

potential solutions (sequences). This process is repeated until either the best

fitness value has been found or the specified maximum number of iterations has

been reached.

182

Initialise Bees Algorithm parameters

Initialise a scout bee population (n)

Generate initial solutions for the component sequencing problem (1st

link) randomly

Generate initial solutions for the feeder arrangement sequencing

problem (2nd

link) randomly

Evaluate the fitness of population

Select best bees (m)

Select sites for neighbourhood search (n-m)

Apply 2-opt Local Search for the 1st link

Apply single-point insertion for 2nd

link

Evaluate the fitness of the neighbours

Select fittest bees from each site

Assign remaining bees to random search ( n- m )

New population of scout bees ( m + ( n – m ) )

Figure 5.6 Flowchart of Bees Algorithm Solving a PCB Problem

183

5.3.3 PCB Component Assembly Using a MBTD Machine

The problem employed to test the ability of the BA to perform PCB assembly

planning for a type–3 machine is detailed in (Leu et al., 1993). For this

problem, the number of components to be placed onto a PCB is 50 and there are

ten feeders, each supplying a different type of component. The coordinates of

the placement positions and the parameters of the assembly machine (speed of

movement of the table, assembly heads and indexing time of the turret)

summarised in Table 5.3, are also given in (Leu et al., 1993).

At first, the PCB problem was solved with the initial positions of the scout bees

chosen completely randomly, as described above. Further improvement to the

Bees Algorithm was then made by introducing a good patch (a good initial

solution) to one of the scout bees. The location of that patch was found by

optimising only the component placement sequence (the first link) thus treating

the problem as a simple TSP.

184

Table 5.3 The Parameters of the MBTD Assembly Machine

Number of components 50

Number of feeders 10

Number of turret heads 2

Indexing time of turret 0.25s/index

Average PCB mounting table speed 60mm/s

Average feeder system speed 60mm/s

Distance between feeders 15mm

185

5.3.4 Experimental Results

The main features of the Bees Algorithm employed in this work have been

chosen by running a set of trail experiments and shown in Table 5.4. As can be

noticed from this table, the number of parameters in the Bees Algorithm has

been reduced from six to three. This is because the elite sites parameter (e) was

not employed and hence the number of recruited bees (nep) around it was not

employed either. In addition, the patch size parameter (ngh) was not needed

because a local search operator e.g. either the 2-Opt or single-point insertion,

was employed for this type of combinatorial problem.

Table 5.4 also shows that the population size n was initialised to 100 bees for

the first iteration. This then dropped to be equal to the number of selected sites

m (20 sites). The recruited bees were then: nep, that is 50 bees for each of the

best m sites.

5.4 DISCUSSION

Figure 5.7 shows the evolution of the best assembly time as the operation of the

Bees Algorithm progresses for the first case where a good starting solution was

not chosen (without “seeding”). After 160 iterations, the best assembly time

was found to be 25.92 seconds. The corresponding component placement

sequence is shown in Figure 5.8.

186

For the second case where seeding was employed, the sequence produced an

assembly time of 29 seconds as shown in Figure 5.9.

Figure 5.10 shows the evolution of the best assembly time and Figure 5.11 the

optimal sequence after 100 iterations with an assembly time of 24.08 seconds.

For comparison, Table 5.5 presents the above results found using the Bees

Algorithm alongside those produced by other researchers for the same problem

using the GA (Ho and Ji, 2007b; Leu et al., 1993; Ong and Tan, 2002) and EP

(Nelson and Wille, 1995).

According to Table 5.5, the performance of the Bees Algorithm with seeding

(24.08 seconds) is superior to that of the simple Bees Algorithm (25.92

seconds), the simple GA (51.5 seconds), the hybrid GA (25.5 second) and EP

(36 seconds) when applied to a benchmark assembly task. In addition, the

number of evaluations needed to perform the optimisation process using the

Bees Algorithm with seeding is less than any of the others.

187

Table 5.4 The Parameters of the Bees Algorithm

Bees Algorithm parameters Symbol

Value

Completely random

initial population

Seeding initial

population with a

good solution

Population size n

100

(1st iteration only)

100

(1st iteration only)

n = m

(for iteration 2

onwards)

n = m

(for iteration 2

onwards)

Number of selected sites m 20 20

Number of recruited bees for best m sites nep 50 50

Number of iterations itr 160 100

188

Figure 5.7 Evolution of Best Assembly Time for Case 1 – Without Seeding

189

Figure 5.8 Optimal Assembly Sequence (25.92 sec) after 160 Iterations for

case 1 – Without Seeding

190

Figure 5.9 Assembly Sequence (29.00 sec) Employed as a Seed for Case 2 –

With Seeding

191

Figure 5.10 Evolution of Best Assembly Time (29.00 sec) for case 2 –

With Seeding

192

Figure 5.11 Optimal Assembly Sequence (24.08 sec) after 100 Iterations for

case 2 – With Seeding

193

Table 5.5 Results Obtained using Different Algorithms

References 1 2 3 4 Bees Algorithm

Optimisation technique GA EP GA HGA No seeding With seeding

Best initial solution (sec) 70 n/a 60 28.83 54.59 29

Number of evaluations 175,000 500,000 50,000 14,150 160,000 100,000

Best assembly time (sec) 51.5 36 26.9 25.5 25.92 24.08

1: (Leu et al., 1993)

2: (Nelson and Wille, 1995)

3: (Ong and Tan, 2002)

4: (Ho et al., 2007b)

194

5.5 SUMMARY

This chapter has described the application of the Bees Algorithm to the

optimisation of pick-up and placement sequences for a PCB assembly machine.

For a benchmark assembly problem, the optimal assembly time of 24.08

seconds obtained by the Bees Algorithm was 6% shorter than the best assembly

time reported in the literature to date, 25.5 seconds. If the assembly machine

was employed continuously, 24 hours a day, 365 days a year, such a saving in

assembly time would represent an annual increase in production of some

72,000 PCBs.

The performance of the Bees Algorithm for solving the component sequencing

and feeder arrangement problems introduced, showed a significant

improvement in the assembly time. However, this reduction in assembly time

achieved by the Bees Algorithm was at the expense of a high number of

candidate solutions that had to be generated and evaluated (as seen with the

number of evaluations presented in Table 5.5).

The computational experiments presented in the chapter have demonstrated that

the Bees Algorithm provides a significant reduction in assembly time compared

to the results obtained using the GA and EP when applied to a benchmark

assembly task.

195

CHAPTER 6

CONCLUSION

This chapter summarises the main contributions of this work and the

conclusions reached. It also provides suggestions for future work.

6.1 CONTRIBUTIONS

This research has introduced the Bees Algorithm as a swarm-based tool for

solving complex optimisation problems.

The specific contributions were:

Employing the Bees Algorithm for tuning its own parameters;

Using the Bees Algorithm for stochastic optimisation problems;

Explaining how the Bees Algorithm can be efficiently applied to

combinatorial optimisation problems, namely, the Travelling Salesman and

Printed Circuit Board problems;

Reducing the number of parameters in the Bees Algorithm from six to

three. This was attained by making some of the parameters dependent on others

and by the selection of the best sites only and not selecting the elite bees as is

196

the case with the standard Bees Algorithm. This in return eliminated the need to

choose the number of recruited bees around the elite sites (nep). In addition, the

patch size parameter was not required because a local search operator e.g. either

the 2-Opt or single-point insertion, was employed for this type of combinatorial

problem.

6.2 CONCLUSIONS

All algorithms have a set of parameters that control their behaviour. These are

employed in order to improve the performance of the algorithm when trying to

arrive at solutions to the problem at hand. This set of parameters depends on the

problem being tackled and its dimension. Unfortunately, these parameters

cannot be generalised to any type of problem of any size.

To arrive at a suitable set of parameters, different algorithms employ different

techniques.

The use of the activity of foraging bees as model for an optimisation process

was shown to provide substantial benefits especially when compared with other

optimisation techniques.

The research in this thesis has extended the application of the Bees Algorithm

to the solution of complex optimisation problems. The new modified Bees

197

Algorithm was employed to optimise stochastic problems based on statistical

analysis. The number of parameters of a standard Bees Algorithm was reduced

from six to three which makes the tuning easier. The algorithm can also adapt

itself to the problem it is dealing with.

When combined with a local search algorithm (2-Opt), the Bees Algorithm was

also able to handle combinatorial problems.

6.3 FUTURE WORK

There are a number of directions that can be pursued in order to enhance the

Bees Algorithm and widen its application potential.

1. The Bees Algorithm could be combined with any of the Ant Colony

Optimisation Algorithms such as Ant Colony System (ACS) to ensure higher

performance and solution quality. For example, a new algorithm could be

developed that employs the Bees Algorithm as a global search and the ACS as a

local search.

2. The Bees Algorithm was employed to tune its own parameters. It can be

tested on the tuning of other algorithms and techniques, for example, online

tuning of the Particle Swarm Optimisation (PSO) algorithm.

198

3. The Bees Algorithm could be applied to other types of combinatorial

optimisation problems, such as Vehicle Routing, Job-Shop Scheduling and

Quadratic Assignment Problems and to other types of assembly machine such

as Pick and Place Machines (PAPs).

4. More statistical analysis could be added to the modified Bees

Algorithm. For example, the standard deviation of the fitness values which

indicates the stability of the solution, a smaller standard deviation meaning a

more stable solution, could be added as part of a multi-objective function to the

modified Bees Algorithm to find the best and most stable solution.

199

REFERENCES

Aarts, E. H. L., Korst, J. H. M. and Vanlaarhoven, P. J. M. 1988. A quantitative

analysis of the simulated annealing algorithm – A case study for the traveling

salesman problem. Journal of Statistical Physics, 50(1-2), pp. 187-206.

Adenso-Diaz, B. and Laguna, M. 2006. Fine-tuning of algorithms using

fractional experimental designs and local search, Operations Research, 54(1):

pp. 99-114.

Ang, M.C., Pham, D.T., Ng, K.W. 2009. Application of the Bees Algorithm

with TRIZ-inspired operators for PCB assembly planning. In: Proceedings of

5th

Virtual International Conference on Intelligent Production Machines and

Systems (IPROMS2006), Cardiff, UK, 2009. pp. 454-459.

Ayob, M., Cowling, P. and Kendall, G. 2002. Optimisation for surface mount

placement machines. In: Proceeding of IEEE ICIT'02. Bangkok, 2002, pp. 498-

503.

Azeem, M.F. and Saad A.M. 2004. Modified Queen Bee Evolution Based

Genetic Algorithm for Tuning of Scaling Factors of Fuzzy Knowledge Base

Controller. IEEE INDICON 2004 Proceedings of the India Annual Conference,

299-303.

200

Bahamish, H. A. A., Abdullah, R. and Abdul Salam, R. A. 2008. Protein

Conformational Search Using Bees Algorithm. In: 2nd IEEE Asia International

Conference on Modeling and Simulation, AICMS 08. Kuala Lumpur, Malaysia:

IEEE Computer Society. pp. 911-916.

Baeck, T., Hoffmeister, F. and Schwefel, H.P. 1991. A survey of evolution

strategies. In: Proceedings of 4th International Conference on Genetic

Algorithm. San Mateo - USA, 1991. Morgan Kaufmann, pp. 2-9.

Bartz-Beielstein, T. and Markon, S. 2004. Tuning search algorithms for real-

world applications: A regression tree based approach. In: Proceedings of

Evolutionary Computation, Portland, OR, USA, 2004, pp. 1111-1118.

Beekman, M., Fathke, R., and Seeley, T. 2006. How does an informed minority

of scouts guide a honey bee swarm as it flies to its new home? Animal

Behavior, 71(1):161–171.

Bilchev, G. and Parmee, I.C. 1995. The ant colony metaphor for searching

continuous design spaces, in Selected Papers from AISB Workshop on

Evolutionary Computing. Springer-Verlag: London. p. 25-39.

Birattari, M., Sützle, T., Paquete, L. and Varrentrapp, K. 2002 A racing

algorithm for configuring metaheuristic. In: Proceedings of Genetic and

Evolutionary Computation Conference. New York, USA, 2002, pp. 11-18.

201

Blackwell, T. and Branke, J., 2004. Multi-swarm optimization in dynamic

environments, Applications of Evolutionary Computing, in Lecture Notes in

Computer Science, Raid,l G.R., Editor, Springer-Verlag: Berlin, Germany.

Bonabeau, E., 1998. Social insect colonies as complex adaptive systems,

Ecosystems, (1), pp. 437-443.

Bonabeau, E., Dorigo, M., and Theraulaz, G. 1999. Swarm intelligence: from

natural to artificial systems. New York: Oxford University Press.

Burke, E.K., Cowling, P.I. and Keuthen, R. 1999. New models and heuristics

for component placement in printed circuit board assembly. In: Proceedings of

ICIIS9, 1999, pp. 133-140.

Camazine, S., Deneubourg, J.L., Franks, N.R., Sneyd, J., Theraula, G., and

Bonabeau, E. 2003. Self-organization in biological systems. Princeton

University Press.

Chandra, B., Karloff, H. and Tovey, C. 1999. New results on the old k-opt

algorithm for the traveling salesman problem. SIAM Journal on Computing.

28(6), pp. 1998-2029.

Coy, S.P., Golden, B.L., Runger, G.C., and Wasil, E.A. 2001. Using

experimental design to find effective parameter settings for heuristics. Journal

of Heuristics, 7 (1): pp. 77-97.

202

Deneubourg, J.L., Aron, S., Goss, S. and Pasteels, J.M., 1990. The self-

organising exploratory pattern of the Argentine ant. Journal of Insect Behaviour

(3), pp. 159-168.

Dorigo, M. and Gambardella, L.M. 1997. Ant colony system: a cooperative

learning approach to the traveling salesman problem. IEEE Transactions on

Evolutionary Computation 1(1): pp. 53-66.

Dorigo, M. and Stützle, T. 2004. Ant colony optimization, Cambridge: MIT

Press.

Dorigo, M., Di Caro, G., and Gambardella, L.M. 1999. Ant algorithms for

discrete optimization. Artificial Life, 5(2), pp. 137-172.

Dorigo, M., Maniezzo V. and Colorni, A. 1996. Ant system: optimization by a

colony of cooperating agents. IEEE Transactions on Systems, Man, and

Cybernetics, Part B 26(1): pp. 29-41.

Dréo, J. and Siarry, P., 2004. Continuous interacting ant colony algorithm

based on dense heterarchy, Future Generation Computer Systems, (20), pp.

841–856.

Eberhart, R., Shi, Y. and Kennedy, J. 2001. Swarm intelligence. San Francisco:

Morgan Kaufmann Publishers.

Engelbrecht, A. P. 2005. Fundamentals of computational swarm intelligence.

203

Hoboken, N.J. Wiley.

Farooq, M 2008. Bee-Inspired Protocol Engineering: From Nature to

Networks, Natural Computing Series, Springer; London, UK.

Figlali, N., Özkale, C., Engin, O. and Figlali, A. 2005. Investigation of ant

system parameter interactions by using design of experiments for job-shop

scheduling problems. In: 35th International Conference on Computers and

Industrial Engineering, Istanbul, Turkey, 2005, pp. 745-750.

Freisleben, B. and Merz, P. 1996. A genetic local search algorithm for solving

symmetric and asymmetric travelling salesman problems. In: Proceedings of

International Conference on Evolutionary Computation. Indianapolis, USA,

1996, pp. 616-621.

Frisch, K.V. 1976. Bees: their vision, chemical senses and language. Revised

Edition ed. Ithaca, N.Y., Cornell University Press.

Fogel, D.B. 2000. Evolutionary computation: toward a new philosophy of

machine intelligence, 2nd edition, New York: IEEE Press.

Fogel, L.J., Owens, A.J., and Walsh, M.J. 1966. Artificial intelligence through

simulated evolution, J. Wiley, New York.

204

Gaertner, D. and Clark, K. 2005. On optimal parameters for ant colony

optimization algorithm. In: Proceedings of the International Conference on

Artificial Intelligence, Las Vegas, Nevada, USA, CSREA, 2005, pp. 83 - 89.

Gambardella, L.M. and Dorigo, M. 1996. Solving symmetric and asymmetric

TSPs by ant colonies. In: Proceedings of IEEE International Conference on

Evolutionary Computation. Nayoya University, Japan, 1996, pp. 622-627.

Garey, M.R. and Johnson, D.S. 1979. Computers and intractability: A guide to

the theory of NP completeness. San Francisco: Freeman.

Ghanbarzadeh, A. 2007. The Bees algorithm. A novel optimisation tool. PhD.

Cardiff University.

Glover, F. and Kochenberger, G. A. 2003. Handbook of metaheuristics,

Boston, MA, USA: Kluwer.

Goldberg, D.E. 1989. Genetic algorithms in search, optimization and machine

learning. Addison-Wesley Longman Publishing Co., MA.

Haddad, O.B., Afshar, A. and Marino, M.A., 2006. Honeybees mating

optimization (HBMO) algorithm: A new heuristic approach for water resources

optimization, Water Resources Management, (20), pp. 661–680.

205

Hao, Z.-F., Cai, R.-C., and Huang, H. 2006. An adaptive parameter control

strategy for ACO. In: Proceedings of International Conference on Machine

Learning and Cybernetics, Dalian, China, 2006, pp. 203-206.

Helsgaun, K, 2000. An effective implementation of the Lin-Kernighan

travelling salesman heuristic. European Journal of Operational Research

126(1), pp. 106-130.

Ho, W. and Ji, P. 2007a. A genetic algorithm to optimise the component

placement process in PCB assembly. International Journal of Advanced

Manufacturing Technology 26, pp. 1397–1401.

Ho, W. and Ji, P. 2007b. Optimal production planning for PCB assembly.

London: Springer.

Holland, J.H. 1975. Adaptation in natural and artificial systems, University of

Michigan Press, Ann Arbor.

Ibaraki, T. 1997. Combination with other optimization methods. Handbook of

evolutionary computation. Ed. Back, T., Fogel, D.B. and Michalewicz, Z. IOP

Publishing Ltd and Oxford University Press.

Jain, A. K. and Dubes, R. C. 1988. Algorithms for Clustering Data. Prentice

Hall, Englewood Cliffs, NJ.

206

Johnson, D.S. and McGeoch, L.A. 1997. Local search in combinatorial

optimization, Ed. Aarts, E. and Lenstra J.K. Wiley, Chichester, p. 215.

Karaboga, D. and Basturk, B., 2008. On the performance of artificial bee

colony (ABC) algorithm, Applied Soft Computing, 8(1), pp. 687-697.

Kennedy, J. and Eberhart, R. 1995. Particle swarm optimization, In:

Proceedings of 1995 IEEE International Conference on Neural Networks, Perth

- Western Australia, 4, 1995, pp. 1942-1948.

Kohonen, T. 1989. Self-Organising and Associative Memory (3rd ed.).

Springer-Verlag ,Berlin.

Kohonen, T. 1990. The Self-Organising Map". Procs. IEEE, 78(9), pp. 1464-

1480.

Knox, J., 1994. Tabu Search Performance on the Symmetric Traveling

Salesman Problem, Computers Ops Res., 21(8), pp. 867-876.

Laporte, G. 1992. The travelling salesman problem: An overview of exact and

approximate algorithms. European Journal of Operational Research 59(2). pp.

231-247.

Lara, C., Flores, J. J. and Calderón, F. 2008. Solving a School Timetabling

Problem Using a Bee Algorithm. Lecture Notes in Computer Science, 2008,

5317, pp. 664-674.

207

Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G. and Shmoys, D.B. 1985. The

traveling salesman problem: a guided tour of combinatorial optimization. John

Wiley and Sons, NY.

Lee, J. Y. and Haj Darwish, A. 2008. Multi-objective Environmental/Economic

Dispatch Using the Bees Algorithm with Weighted Sum. In: EU-Korea

Conference on Science and Technology (EKC2008). Heidelberg, Germany:

Springer. pp. 267-274.

Leipala, T. and Nevalainen, O. 1989. Optimisation of the movements of a

component placement machine. European Journal of Operational Research.

38, pp. 167-177.

Leu, M.C., Wong, H. and Ji, Z. 1993. Planning of component

placement/insertion sequence and feeder setup in PCB assembly using genetic

algorithm. ASME. Trans. Journal of Electronic Packaging. 115, pp. 424-432.

Lin, S. and Kernighan, B.W. 1973. An effective heuristic algorithm for the

traveling-salesman problem. Operations Research. 21(2), pp. 498-516.

Ling, W. and Luo, H. 2007. An adaptive parameter control strategy for ant

colony optimization. In: Proceedings of International Conference on

Computational Intelligence and Security. Harbin - China, 2007, pp. 142-146.

208

Lucic, P. and Teodorovic, D. 2003. Computing with bees: attacking complex

transportation engineering problems. International Journal on Artificial

Intelligence Tools. 12(3), pp. 375-394.

Maimon, O.Z. and Brha, D. 1998. A genetic algorithm approach to scheduling

PCBs on a single machine. International Journal of Production Research 36(3):

pp. 761- 784.

Martens, D., De Backer, M., Haesen, R., Vanthienen, J., Snoeck, M., Baesens,

B. 2007. Classification with Ant Colony Optimization", IEEE Transactions on

Evolutionary Computation, 11(5), pp. 651-665.

Mathur, M., Karale, S. B., Priye, S., Jayaraman, V. K. and Kulkarni, B. D.

2000. Ant colony approach to continuous function optimization. Ind. Eng.

Chem. Res. 39(10), pp. 3814-3822.

Merz, P. and Freisleben, B. 1997. Genetic local search for the TSP: New

results. In: Proceedings of the 1997 IEEE International Conference on

Evolutionary Computation. Indianapolis, USA, 1997, pp. 159-164.

Michalewicz, Z. 1996. Genetic algorithms + data structures = evolution

programs. 3rd rev. and extended ed. Berlin: Springer-Verlag.

209

Nelson, K.M. and Wille, L.T. 1995. Comparative study of heuristics for optimal

printed circuit board assembly. In: Proceedings of Southcon 95. Fort

Lauderdale, FL, USA, 1995, pp. 322 - 327.

Okano, H., Misono, S. and Iwano, K. 1999. TSP construction heuristics and

their relationships to the 2-Opt. Journal of Heuristics (1), pp. 71-88, Kluwer

Academic Publishers, Boston.

Oliver, I. M., Smith, D.J. and Holland, J.R.C. 1987. A study of permutation

crossover operators on the traveling salesman problem. In: Proceedings of

Second Int. Conf. Genetic Algorithms and their Applications. Cambridge, MA,

USA, 1987, pp. 224-230.

Omran, M. G., Engelbrecht, A. P. and Salman, A. 2005. Particle swarm

optimization method for image clustering. Int. Journal on Pattern Recognition

and Artificial Intelligence 19(3), p. 297-322.

Ong, N. and Khoo, L.P. 1999. Genetic algorithm approach in PCB assembly.

Integrated Manufacturing Systems, 10(5): pp. 256-265.

Ong, N.S. and Tan, W.C. 2002. Sequence placement planning for high-speed

PCB assembly machine. Integrated Manufacturing Systems. 13 (1), pp. 35-46.

Passino, K.M., 2002. Biomimicry of bacterial foraging for distributed

optimisation and control, IEEE Control Systems Magazine (June), pp. 52-67.

210

Passino, K.M., Seeley, T.D., and Visscher, P.K., 2008. Swarm cognition in

honeybees, Behavioral Ecology Sociobiology, 62(3), pp. 401-414.

Pellegrini P., Favaretto D. and Moretti E. 2006. On MAX -MIN ant system's

parameters, in Ant Colony Optimization and Swarm Intelligence. pp. 203-214.

Pham, D.T., Afify, A.A. and Koç, E. 2007b. Manufacturing cell formation

using the Bees Algorithm. In: 3rd International Virtual Conference on

Intelligent Production Machines and Systems (IPROMS 2007), Whittles,

Dunbeath, Scotland, 2007, pp. 523-528.

Pham, D.T. and Oztemel, E. 1992. Control Chart Recognition Using Neural

Networks. Journal of Systems Engineering. pp. 256-262.

Pham, D.T. and Sholedolu, M. 2008. Using a hybrid PSO-Bees Algorithm to

train Neural Networks for Wood Defect Classification. In: 4th International

Virtual Conference on Intelligent Production Machines and Systems, IPROMS.

Pham, D.T., Ghanbarzadeh, A., Koç, E., Otri, S., Rahim, S. and Zaidi, M.

2006b. The Bees Algorithm - A novel tool for complex optimisation problems.

In: Proceedings of 2nd Virtual International Conference on Intelligent

Production Machines and Systems (IPROMS2006), Cardiff, UK, 2006. pp.

454-459.

211

Pham, D.T., Koc, E., Ghanbarzadeh, A., and Otri, S. 2006c. Optimisation of the

weights of multi-layered perceptrons using the Bees Algorithm, in 5th

International Symposium on Intelligent Manufacturing Systems, Sakarya,

Turkey, 2006.

Pham, D. T., Koc, E., Lee, J.Y. and Phrueksanant, J., 2007c. Using the Bees

Algorithm to schedule jobs for a machine. In: LAMDAMAP, 8th international

Conference on Laser Metrology, CMM and Machine Tool Performance,

Cardiff, Euspen, UK, 2007, pp. 430-439.

Pham D. T., Lee J. Y., Haj Darwish A., and Soroka A. J. 2008a. Multi-objective

Environmental/Economic Power Dispatch using the Bees Algorithm with

Pareto optimality. In: 4th International Virtual Conference on Intelligent

Production Machines and Systems (IPROMS 2008a), Whittles, Dunbeath,

Scotland, 2008.

Pham, D.T., Castellani, M. and Fahmy, A.A., 2008e. Learning the inverse

kinematics of a robot manipulator using the Bees Algorithm. In: Proceedings of

INDIN, 2008, pp. 493-498.

Pham, D. T., Castellani, M., Sholedolu, M. and Ghanbarzadeh, A. 2008c. The

Bees Algorithm and Mechanical Design Optimisation. pp250-255

http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/c/Castellani:M=.html

http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/s/Sholedolu:M=.html

http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/g/Ghanbarzadeh:A=.html

212

Pham, D.T., and Karaboga, D., 2000. Intelligent optimisation techniques:

Genetic Algorithms, Tabu Search, Simulated Annealing and Neural Networks.

Springer-Verlag, London, UK.

Pham, D.T. and Castellani, M. 2009. The Bees Algorithm – Modelling

Foraging Behaviour to Solve Continuous Optimisation Problems. Proc.

ImechE, Part C, 223(12): 2919-2938.

Pham, D. T. and Liu, X. 1995. Neural Networks for Identification, Prediction

and Control. London: Springer.

Pham D. T. and Sahran S., 2006. “Control Chart Pattern Recognition with

Spiking Neural Networks”, Proceedings of the Second Virtual International

Conference on Intelligent Production Machines and Systems, D. T. Pham, E. E.

Eldukhri and A. J. Soroka (eds), Elsevier (Oxford), 319-325, ISBN 0-08-

045157-8.

Pham, D.T., Castellani, M. and Ghanbarzadeh, A., 2007e. Preliminary design

using the Bees Algorithm. In: Proceedings of Eighth LAMDAMAP

International Conference on Laser Metrology, CMM and Machine Tool

Performance, Cardiff, UK, 2007, pp. 420-429.

Pham, D.T., Ghanbarzadeh, A., Koc, E., and Otri. S., 2006d. Application of the

Bees Algorithm to the training of radial basis function networks for control

213

chart pattern recognition. In: 5th CIRP International Seminar on Intelligent

Computation in Manufacturing Engineering, CIRP ICME, 2006, Ischia, Italy.

Pham, D.T., Ghanbarzadeh, A., Koc, E., Otri, S., Rahim, S., and Zaidi, M.,

2005. The Bees Algorithm. Technical Note, Manufacturing Engineering Centre,

Cardiff University, UK.

Pham, D. T., Muhamad, Z., Mahmuddin, M., Ghanbarzadeh, A., Koc, E. and

Otri, S. 2007a. Using the Bees Algorithm to optimise a support vector machine

for wood defect classification. IPROMS 2007 Innovative Production Machines

and Systems Virtual Conference, Cardiff, UK.

Pham, D. T., Pham, Q. T., Ghanbarzadeh, A., and Castellani, M. 2008b.

Dynamic Optimisation of Chemical Engineering Processes Using the Bees

Algorithm. In: 17th IFAC World Congress COEX, South Korea, 2008, pp.

6100-6105.

Pham, D.T., Otri, S. and Darwish, A.H., 2007d. Application of the Bees

Algorithm to PCB assembly optimisation. In: IPROMS, 3rd International

Virtual Conference on Intelligent Production Machines and Systems. Whittles,

Dunbeath, Scotland. 2007, pp. 511-516.

Pham, D.T., Soroka, A.J., Ghanbarzadeh, A., Koc, E., Otri, S. and

Packianather, M.,2006e. Optimising neural networks for identification of wood

214

defects using the Bees Algorithm. In: IEEE International Conference on

Industrial Informatics, Singapore, 2006, pp. 1346-1351.

Pham, D.T., Otri, S., Afify, A., Mahmuddin, M., and Al-Jabbouli, H. 2007f.

Data clustering using the Bees Algorithm. In: 40th CIRP International

Manufacturing Systems Seminar Liverpool, UK, 2007.

Pham, D. T., Otri, S., Ghanbarzadeh, A., and Koc, E. 2006a. Application of the

Bees Algorithm to the Training of Learning Vector Quantisation Networks for

Control Chart Pattern Recognition," in 2nd IEEE International Conference on

Information and Communication Technologies: From Theory to Applications.

Damascus, Syria, 2006, (1), pp. 1624 -1629.

Pilat, M.L. and White, T. 2002. Using Genetic Algorithms to optimize ACS-

TSP. In: Proceedings of the Third International Workshop on Ant Algorithms.

Springer-Verlag, 2002, pp. 282-287.

Rechenberg, I. 1965. Cybernetic solution path of an experimental problem,

Library Translation no. 1122. Ministry of Aviation, Royal Aircraft

Establishment, Farnborough, Hants UK.

Rego, C. and Glover, F. 2002. Local search and metaheuristics. Gutin and

Punnen, pp. 309 - 368.

215

Rosenkrantz, D.J., Stearns, R.E. and Lewis, P.M. 1977. An analysis of several

heuristics for the travelling salesman problem. SIAM Journal on Computing

6(2), pp. 563-581.

Saad, E.M., Awadalla, M.H., Darwish, R.R. 2008"A Data Gathering Algorithm

for a Mobile Sink in Large-Scale Sensor Networks," Wireless and Mobile

Communications, 2008. ICWMC '08. The Fourth International Conference on,

vol., no., pp.207-213.

Sadik, S., Ali, A., Ahmad, F. and Suguri, H. 2006. "Using Honey Bee

Teamwork Strategy in Software Agents," Computer Supported Cooperative

Work in Design, 2006. CSCWD '06. 10th International Conference on, pp.1-6.

Sato, T., and Hagiwara, M. 1997. Bee System: finding solution by a

concentrated search. In: Proceedings of IEEE International Conference on

Systems, Man, and Cybernetics, USA, 1997, pp. 3954-3959.

Seeley, T.D. 1996. The wisdom of the hive: The social physiology of honey bee

colonies, Cambridge, Massachusetts: Harvard University Press.

Seeley, T. D. and Visscher, P. K. 2003. Choosing a home: how the scouts in a

honey bee swarm perceive the completion of their group decision making.

Behav Ecol Sociobiol, (54) pp. 511–520.

216

Socha K. 2003. The influence of run-time limits on choosing ant system

parameters. In: Proceedings of Genetic and Evolutionary Computation

Conference (GECCO). Chicago, USA, 2003 (2723), pp. 49-60.

Solnon, C. 2002. Boosting ACO with a preprocessing step. Applications of

Evolutionary Computing, Kinsale, Ireland, pp. 41-50.

Stützle, T., 1997. MAX-MIN Ant System for the quadratic assignment

problem. Technical Report, AIDA-97-4, FB Informatik, TU Darmstadt,

Germany, 1997.

Stützle, T. and Hoos, H. 1997. MAX-MIN ant system and local search for the

traveling salesman problem. In: Proceedings of ICEC'97 - 1997 IEEE 4th

International Conference on Evolutionary Computation. Indianapolis, USA,

1997, pp. 308-313.

Sung, H. J. 2003. Queen-Bee Evolution for Genetic Algorithms. Electronic

Letters, 39(6), 575- 576.

Teodorovic, D., Lucic, P., Markovic, G. and Orco, M. D. 2006. Bee colony

optimization: principles and Applications, Neural Network Applications in

Electrical Engineering, pp. 151–156.

217

Tereshko, V. and Lee, T., 2002. How information-mapping patterns determine

foraging behaviour of a honey bee colony, Open Systems and Information

Dynamics, 9(2), pp. 181-193.

Tovey, C. A. 2004. The honey bee algorithm, a biologically inspired approach

to internet server optimization. Engineering Enterprise. pp.13-15.

Watanabe, I. and Matsui, S. 2003. Improving the performance of ACO

algorithms by adaptive control of candidate set. In: Proceedings of

Evolutionary Computation Conference, Chicago, IL, USA, 2003 (2), pp. 1355-

1362.

Wedde, H.F., Farooq, M. and Zhang Y. 2004. BeeHive: An efficient fault-

tolerant routing algorithm inspired by honey bee behavior. 2nd

ed. Lecture

Notes in Computer Science. 83-94.

Wedde, H.F., Farooq, M., Pannenbaecker, T., Vogel, B., Mueller, C., Meth, J.,

and Jeruschkat, R. 2005. BeeAdHoc: An energy efficient routing algorithm for

mobile ad hoc networks inspired by bee behaviour. In: Proceedings of the 2005

Conference on Genetic and Evolutionary Computation. ACM Press:

Washington DC, USA, 2005, p. 153-160.

218

White, C.M. and Yen, G.G. 2004. A hybrid evolutionary algorithm for

travelling salesman problem. In: Proceedings of Congress on Evolutionary

Computation, CEC, 2004. pp. 1473-1478.

Wiener, R. 2003. Branch and bound implementations for the travelling

salesperson problem - Part 1. Journal of Object Technology 2(2), pp. 65-86.

Wolpert, D.H. and Macready, W.G., 1997. No free lunch theorems for

optimization, IEEE Transactions on Evolutionary Computation, 1(1), pp.:67-

82.

Wong, H. and Leu, M.C. 1993. Adaptive genetic algorithm for optimal printed

circuit board assembly planning. Annals of the CIRP. 42, pp. 17-20.

Wong, K. Y., Komarudin, 2008. Parameter tuning for ant colony optimization:

A review. In: Proceedings of the International Conference on Computer and

Communication Engineering. Kuala Lumpur - Malaysia, 2008. pp. 542-545.

Yang, X.S. 2005. Engineering optimizations via nature-Inspired virtual Bee

Algorithms. In: IWINAC (2). Springer. p. 317-323.

Yeo, S.H., Low, C.W. and Yong, K.H. 1996. A rule-based frame system for

concurrent assembly machines. The International Journal of Advanced

Manufacturing Technology. 12, pp. 370-376.

219

Zecchin, A.C., Simpson, A.R., Maier, H.R. and Nixon, J.B. 2005. Parametric

study for an ant algorithm applied to water distribution system optimization.

IEEE Transactions on Evolutionary Computation. 9(2): pp. 175-191.

220

Documents

IMPROVING THE BEES ALGORITHM FOR COMPLEX … · Algorithm to solve a stochastic optimisation problem. The new modified Bees Algorithm was tested on the optimisation of parameter values